Superconductor-insulator transition and Bose-Einstein condensate - Bardeen-Cooper-Schrieffer crossover in the Rashba Moat Band

We study the superconducting transition in a two-dimensional electron gas with strong Rashba spin-orbit coupling. We assume low electron density, such that only the majority spin band participates in the transition. We show that the superconducting transition follows either the Bose-Einstein condensation (BEC), or the Bardeen-Cooper-Schrieffer (BCS) scenarios, depending on the position of the chemical potential with respect to the bottom of the majority band, and the strength of the Coulomb repulsion between electrons. Hence, the BEC-BCS crossover in this system can be driven either by the change in the chemical potential, or the distance to a gate. Following the path integral approach, we also study the behavior of the system at finite temperature and in particular evaluate the effects of fluctuations, which is especially important in our two-dimensional model.

SUPERCONDUCTOR-INSULATOR TRANSITION AND BOSE-EINSTEIN CONDENSATE - BARDEENCOOPER-SCHRIEFFER CROSSOVER IN THE RASHBA MOAT BAND by Hassan Allami A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah August 2019 Copyright c Hassan Allami 2019 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Hassan Allami has been approved by the following supervisory committee members: Oleg Starykh , Chair(s) 5/6/2019 Date Approved Dima Pesin , Member 5/6/2019 Date Approved Eugene Mishchenko , Member 5/6/2019 Date Approved Vikram Deshpande , Member 5/6/2019 Date Approved Firas Rassoul-Agha , Member 5/6/2019 Date Approved by Peter Trapa , Chair/Dean of the Department/College/School of Physics and Astronomy and by David B. Kieda , Dean of The Graduate School. ABSTRACT We study the superconducting transition in a two-dimensional electron gas with strong Rashba spin-orbit coupling. We assume low electron density, such that only the majority spin band participates in the transition. We show that the superconducting transition follows either the Bose-Einstein condensation (BEC), or the Bardeen-Cooper-Schrieffer (BCS) scenarios, depending on the position of the chemical potential with respect to the bottom of the majority band, and the strength of the Coulomb repulsion between electrons. Hence, the BEC-BCS crossover in this system can be driven either by the change in the chemical potential, or the distance to a gate. Following the path integral approach, we also study the behavior of the system at finite temperature and in particular evaluate the effects of fluctuations, which is especially important in our two-dimensional model. for fun and misery CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTERS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. DEFINING AND FORMULATING THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Single particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Projected interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Attractive Cooper channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Long-range Coulomb channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. SINGLE PAIR WAVEFUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Without Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effects of Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Strong screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Weak screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 28 29 30 31 VARIATIONAL BCS WAVEFUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Without Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effects of Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Strong screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Weak screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 22 23 24 25 41 45 48 50 ACTION TREATMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Building the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Making the effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Without Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Effects of Coulomb repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Fluctuations of the superconducting field . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Without fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Fluctuations of the superconducting field . . . . . . . . . . . . . . . . . . . . . . . . . 57 58 60 63 63 65 68 74 75 77 6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 vi LIST OF FIGURES 1.1 1.2 Spin-orbit coupling in 2DEG. a: Schematic of a GaAs heterostructure. A two-dimensional electron gas (2DEG) forms between the GaAs layer and the layer with donor atoms (such as Aluminum). As a result of the broken inversion symmetry, an electric field (the yellow arrows) develops in the 2DEG. b: In the laboratory frame, there is only an electric field perpendicular to the 2DEG while the itinerant electrons (or holes) are moving. c: In the moving electron’s (or hole) reference frame, according to special relativity, there will be a magnetic field perpendicular to the electric field and the electron’s direction of motion (or more accurately its momentum). . . . . . . . . . . . . . . . . . . . 3 Schematic view of the spin precession of an electron moving in a 2D layer with a SOC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Schematic view of BCS-BEC crossover. In the BCS limit, pairs are strongly overlapping while in the BEC limit, pairs are molecular-like and their size is smaller than the interparticle distance. In the intermediate state, the pair size is comparable with the interparticle distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1 Schematic for the gated 2D layer system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Main: Side-view of the two 2D Rashba spin-orbit split bands for Λ > 0 situation. The insert shows the top view of the ring-like Fermi surface for Λ + Φ < 0 case; see Eq. (4.22) and discussion related to it. . . . . . . . . . . . . . . . . . 23 3.1 Comparing numerical and analytic d-dependence of Coulomb correction to gc (Eq. (3.10)) for Ec = 10ER and for three different Λ’s in the range dp0 1. . 32 3.2 The numerical solution of the Schrödinger equation (3.7) for Ec = ER /10 and dp0 = 1000 and two incrementally successive values of g. The numerical value of g/κ is printed in each panel. It can be seen how the bound state emerges with a finite size in the lower panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Comparing numerical and analytical (Eq. (3.26)) Λ-dependence of gc in noscreening case and for Ec = ER . The dashed curve shows the result in the absence of Coulomb repulsion (Eq. (3.5)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Critical gcb for the two-particle bound state formation in various regimes of screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Phase diagram in the absence of Coulomb repulsion. gc line on the Λ positive side is drawn based on Eq. (3.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 ˜ (Λ̃). Two important points of the function are shown The universal function ∆ in the figure: A is the point where the chemical potential touches the band ˜ 0] ≈ edge, which is the point of the BEC-BCS crossover, with coordinates [Λ̃0 , ∆ ˜ [0, 1.393]; B is the point where the maximum of ∆ is located, with coordinates [Λ̃m , ∆˜ m ] ≈ [−1.424, 1.655]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Phase diagram in the presence of Coulomb repulsion. The boundaries were found numerically for a weakly screened Coulomb case with parameters: dp0 = 1000 and Ec = ER /10. Λeff on the x-axis corresponds to the value of Λ + Φ in the normal state, which is the same as bare Λ on the insulator side and is renormalized strongly on the metallic side according to Eq. (4.21). . 46 4.4 Comparing numerical solution of Eq. (4.8) (labeled as iteration) and the ansatz (4.16) (labeled as ansatz) for Λ = 0 (top panel) and Λ = ER /100 (bottom panel). g is chosen to be slightly above gc , such that the diluteness condition (4.20) holds. For the top panel, g/κ = 0.175, and for the bottom panel, g/κ = 0.52. In both cases, screening is very weak: dp0 = 1000 and Ec = ER /10. In order to make the ansatz solution, the Schrödinger equation (3.7) has been solved numerically to find α p and then using Eq. (4.17) and (4.13) the normalization constant δ was found. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5 Typical ∆ p numerical solution of Eq. (4.8) for Λ = − ER and very weakly screened Coulomb: dp0 = 1000 and Ec = ER /10. The lowest panel is a typical solution of the BCS kind while the top one was found in the BEC region. The middle panel represents the boundary case where Λ + Φ = 0. For each panel, numerical values of g and Λ + Φ are written on the plot. . . . . . . . . . . . . . . . . . . 55 4.6 Φ p numerical solution of Eq. (4.8) for Λ = − ER and very weakly screened Coulomb: dp0 = 1000 and Ec = ER /10. Each curve is labeled by its corresponding g/κ value, while the same set of g is used as in Figure 4.5 . . . . . . . . . 56 5.1 The universal function T̃c (Λ̃). Two important points of the function are shown in the figure: A is when the chemical potential touches the band edge where [Λ̃0 , T̃c,0 ] ≈ [0, 0.736] and B is the maximum Tc point where [Λ̃m,Tc , T̃c,m ] ≈ [−1.784, 0.888] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 The condensate ∆0 spontaneously chooses its phase to be zero, ∠∆0 = 0. Then P and A are the fluctuations in the phase and the amplitude of the order parameter, ∆, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 viii CHAPTER 1 INTRODUCTION This dissertation is studying a problem of superconductivity in a two-dimensional system with strong Rashba spin-orbit coupling (SOC). Before formulating the problem and starting to analyze it, we would like to provide a background by giving general information about SOC in condensed matter systems and a brief introduction on relevant topics in superconductivity. We shall also try to review the existing literature related to the present work. As we will discuss in details in the coming chapters, the model that we will build accommodates two different regimes of superconductivity, known as the Bose-Einstein Condensate (BEC) regime, and the standard regime, which is named after Bardeen, Cooper and Schrieffer (BCS) for their seminal work [10] in introducing a microscopic theory of superconductivity. The smooth transition between these two regimes is known as the BEC-BCS crossover. The other important aspect of our model is the special importance of the electrostatic Coulomb repulsion. Finally, we shall not forget about the critical importance of the fact that the system under our investigation will be a two-dimensional system which brings about critical effects to superconducting systems, especially concerning the fluctuations in the system. Therefore, in the second part of this chapter, we are going to focus more on the above topics in superconductivity. Since it is the spin-orbit coupling that prepares the playground of our model, let us start by introducing SOC and briefly explore different aspects and different manifestations of it in the first section of this chapter. To put our work into context, we will also review the relevant literature. 1.1 Spin-orbit coupling If we can count electron’s spin as essentially a relativistic property of electrons, as it is the direct consequence of Dirac’s equation [49], then spin-orbit coupling (SOC) is a 2 combination of two relativistic phenomena. In the nonrelativistic limit, Dirac’s equation reduces to Schrödinger equation. Then expanding around the non-relativistic limit, among the first corrections is the SOC term that couples the electron’s spin to its momentum and the gradient of the external potential. The term gets its name from the atomic context, where the nucleus generates the potential gradient and the momentum manifests itself in the orbital motion, resulting in a correction term in the Hamiltonian of Hydrogen-like atom proportional to L · S, the result of which can be seen in the splitting of spectral lines. In the context of atomic physics, the SOC correction to the energy levels is usually of the same order of the relativistic correction to the kinetic energy and the quantum mechanical fluctuations of a relativistic particle know as zitterbewegung. All three terms can be derived from a proper expansion of the Dirac equation in the non-relativistic limit. The three terms together describe the fine splitting of spectral lines known as fine structure (not to be confused with the smaller hyperfine structure correction that originates in the interaction of the nucleus magnetic moment and the magnetic field produced by the orbiting electron). The splitting was observed before the invention of Quantum Mechanics in 1887 by Albert Michelson and Edward Morley [114]. Schrödinger proposed the zitterbewegung term in 1930 [160], and in 1940, Sommerfeld completed the proper theoretical explanation where he also introduced fine-structure constant [163]. To develop an intuitive picture, one can think of SOC as a momentum-dependent Zeeman effect, which states that the energy contribution of a particle with a magnetic moment µ in a magnetic field B is given by HZ = −µ · B. The next step would be to realize that when an electron is moving in electrical potential with gradient, i.e., in an electric field, due to special relativity, there will be a magnetic field in the electron’s reference frame proportional to −v × E. In the quantum mechanical setting, the momentum operator p replaces v to construct the momentum-dependent magnetic field and hence the momentum-dependent Zeeman effects or in other words, the spin-orbit coupling. Figure 1.1 presents this simple picture graphically for the case of a two-dimensional electron gas. In the context of condensed matter physics and band structure of crystals, it was first Elliott [60] and Dresselhaus et al. [54] who considered the importance of SOC. Elliot [60] specifically pointed out how the presence of inversion symmetry leads to at least double 3 Figure 1.1. Spin-orbit coupling in 2DEG. a: Schematic of a GaAs heterostructure. A two-dimensional electron gas (2DEG) forms between the GaAs layer and the layer with donor atoms (such as Aluminum). As a result of the broken inversion symmetry, an electric field (the yellow arrows) develops in the 2DEG. b: In the laboratory frame, there is only an electric field perpendicular to the 2DEG while the itinerant electrons (or holes) are moving. c: In the moving electron’s (or hole) reference frame, according to special relativity, there will be a magnetic field perpendicular to the electric field and the electron’s direction of motion (or more accurately its momentum). degeneracy of bands. And as one of the first detectable consequences of SOC in crystals, Dresselhaus [55] showed that it could change the cyclotron resonance in silicon and germanium. Then in another work [53] using group theory, Dresselhaus demonstrated that in a zinc-blende crystal structure that lacks inversion symmetry, the SOC causes spin splitting in the band structure, an anisotropic and cubic in momentum correction that is now known as cubic Dresselhaus spin-orbit term. At the same time, similar work was published by Parmenter [130]. It’s noteworthy to mention that zinc-blende is the crystal structure of many semiconductors such as GaAs, InSb, and CdTe. Another crystal structure that is very common in semiconductors is wurtzite structure, which is also inversion asymmetric. It is the crystal structure of semiconductors such as GaN, CdS, and ZnO. A few years later after the works of Dresselhaus on the symmetry properties of zincblende, in 1959 Rashba, published two works [136, 138] on the effects of symmetry properties of wurtzite structure on electronic band structure. The second paper [138], which was written jointly with Sheka, was considering the effects of spin-orbit coupling (or interaction as they called it) as well. In that second paper, they showed that as a result of SOC, the energy dispersion of s-electrons near the Γ point of Brillouin zone (that is near k = 0) becomes isotropic and linear in k, which leads to the emergence of a 4 ring of minimum at some finite k (or maximum if it’s a hole band) in one branch of the band structure. This is the crucial feature of what is called Rashba moat band and is the signature of Rashba SOC. Around the same time, similar works to that of Rashba and Sheka appeared that were studying band structure of wurtzite crystals [9, 35, 36, 74]. However, Rashba’s work was not only more comprehensive than others, he in collaboration with Boiko and Sheka also followed up the work, in a series of papers, by studying detectable consequences of such a peculiar band structure [24, 137, 141, 142]. Moreover, in a couple of other works, they also analyzed the consequences of the spin-orbit term in the zincblende structure [23, 139, 140]. In the following years, the predicted phenomena were observed experimentally [16, 111], though some of them as late as ’80s [51]. In the coming years in the ’70s, quasi-two-dimensional-semiconductor systems received a lot of attention. In 1971, Antcliffe et al. [5] observed beating in Shubnikov-de Haas oscillation of n-type layers of HgCdTe, which was a few years later explained by the linear spin splitting of Rashba-type [87]. Although such a beating pattern can have different sources, it is now considered as a signature for the presence of a linear 2D spin splitting. While in the ’70s there were still not many experiments around motivating theoretical studies by demonstrating peculiar phenomena that 2D systems with SOC can exhibit, in 1979, Vasko and Prima studied the possible effects that such systems can manifest [175, 176], including the induced spin-polarization by external electric field, a phenomena that is now known as Edelstein effect [58]. As Rashba in a memorial article puts it [143]: “There are such instances in the development of science when experiment becomes not only ripe to absorb concepts already existing in the theory but directly requires them.” In 1983, two major experimental works on GaAs heterostructure came out (for a schematic view of the heterostructure see Figure 1.1). The Stein et al. work [164] was about electron spin resonance in the system, suggesting that some spin-splitting persists down to the limit of zero external magnetic field. Studying magnetoresistivity of the hole system, Störmer et al. observed beatings in the Shubnikov-de Haas oscillation signaling the spin-splitting in the hole bands of the heterostructure as well [165]. Bychkov and Rashba realized how both of the experiments could be explained using the same SOC model that was developed to describe the wurtzite band structure. The unifying explana- 5 tory power of their famous 1984 works [31, 32] immediately attracted a lot of attention and it triggered extensive, multifaceted studies that continue to the present day. The discovery of giant magnetoresistance (GMR) in 1988 [21] following by development of spin valve [48] officially started the field of spintronics, a field that tries to harness the spin degree of freedom of charge carriers to implement useful devices. A simple GMR system is made of two layers of ferromagnetic material that sandwich a thin (of the order of a nanometer) layer of a metal. When the two magnets have parallel magnetization, electric current can pass through the device with low resistivity (the gate is open), but when they are anti-parallel, the resistivity significantly increases, which results in the significant drop of the electric current (the gate is closed). The physics behind this effect originates in the fact that in such a setting electrons with up and down spins scatter differently [33]. Sometime after the original discovery of GMR, it turned out that if one replaces the thin metallic layer with an insulator, similar and even a more significant effect would be present, now know as tunneling magnetoresistance (TMR) [116]. Today many of magnetic memory technologies such as RAMs and hard disk heads are based on this physics. The Physics Nobel prize of 2007 was jointly given to Albert Fert and Peter Grünberg for their works on the GMR effect. Pretty much since the beginning of the development of spintronics, researchers tried to utilize SOC-based spin splitting to design novel materials and new devices. In 1990, Datta and Das proposed [43] spin field effect transistor (spin-FET) that combines GMR setting with the peculiar property of spin-orbit semiconductors. We know that generally, a magnetic moment in an external magnetic field would precess around the external field, a classical dynamics that has its counterpart in Quantum Mechanics. The precession frequency is known as Larmor frequency, and it’s proportional to the magnitude of the magnetic field. A similar phenomenon happens to the spin of a moving electron in a material with SOC. In such systems, the moving particle observes a momentum-dependent magnetic field, and consequently, its spin precesses trying to align itself with the observed magnetic field. Figure 1.2 shows a schematic view of this situation. The spin-FET device uses such a layer sandwiched between two magnets. The idea is to utilize the spin precession in the middle bit to turn the valve on and off instead of flipping the magnets orientations. This relies on the tunability of SOC magnitude in heterostructure. That is because the strength of SOC 6 Figure 1.2. Schematic view of the spin precession of an electron moving in a 2D layer with a SOC. in a heterostructure depends on the shape of the electrical potential’s gradient that is in principle controllable by a gate. By controlling the strength of SOC, one can control the precession time to get the desired result out of the device. The tunability of the SOC has been experimentally shown in the ’90s for different heterostructure [61, 122, 161]. However, due to technical difficulties, such as conductance mismatch between the metallic magnets and the semiconductor layer, it took a while until a functional implementation of the device was realized [38, 99]. Since the Datta-Das transistor, there have been a variety of efforts to utilize SOC for spintronics purposes. Here below we provide a summary of some of the major categories. In 1971, Dyakonov and Perel [56] showed that carries with different spin scattering off of impurities with SOC diffuse in opposite directions. This results in a pure spin flow perpendicular to the electric current, a phenomenon that is known as spin Hall effect. While Dyakonov-Perel mechanism is extrinsic to the host material, later after the spintronics revolution, it was shown that the same phenomenon could occur in semiconductor systems with intrinsic spin-orbit band splitting [119, 162]. Sinova et al. [162] also showed that ballistic spin conductivity associated with this Hall current would have a universal value connected to a topological feature of the whole band in the Brillouin zone known as Berry phase. The spin Hall effect (SHE) was experimentally observed in GaAs systems [93, 181], though it’s not certainly established that it’s due to the extrinsic or intrinsic mechanism. Also in an attempt to combine spin transistor and SHE, a spin Hall effect transistor has been implemented by Wunderlich et al. [182]. In 2006, Valenzuela and Tinkham reported the SHE in a diffusive metallic conductor as well [173]. The fruitful controllability of the strength of SOC in heterostructure systems makes it possible to generate a gradient in the apparent spin-orbit magnetic field in the 2DEG by proper gating. As an alternative technique to SHE, providing such an inhomogeneity 7 in the momentum-dependent magnetic field, one can construct an equivalent of a SternGerlach setup to generate spin polarization. Implementing this method, Ohe et al. [126] reported nearly full spin-polarization. Unlike SHE systems in which the polarized spins are perpendicular to the current plane, in the Stern-Gerlach spin filter, the polarization orientation stays in the plane. While in SHE or Stern-Gerlach spin filter systems flow of electric current leads to spin polarization, the spin galvanic effect does the opposite. That is to say, generating a nonequilibrium spin density in a system with SOC would induce an electric current perpendicular to the spin polarization direction and the direction of broken inversion symmetry [180]. This phenomenon was first experimentally observed in quantum wells [67] and later was also implemented in a NiFe/Ag/BiAg device in which the NiFe pumps the spin density and SOC occurs at the Ag/BiAg interface [153]. The Edelstein effect that we mentioned above can be understood as the Onsager reciprocal of this effect and has also been observed in quantum wells [68] and in strained semiconductors [92]. Finally, we should mention the importance of SOC in the realization of many of the so-called topological phases of matter. In particular, it plays an essential role in topological insulators (TI) and topological superconductors. Although quantum spin Hall effect (QSHE), which is the basis of topological insulators, was first proposed to occur in graphene [91], it was first discovered in HgTe/CdTe quantum well [98], a classic SOC heterostructure. Its existence was predicted a year before by the now classic paper of Bernevig et al. [20]. Later QSHE was reported to exist in InAs/GaSb quantum well as well [96], another heterostructure with strong SOC. TIs are insulators in bulk, but they support two conducting states on the edge that are propagating in the opposite directions, which is not that special. It is the existence of SOC that makes the TIs edge states special by locking the propagating direction to the spin polarization. The spin-momentum lock effectively suppresses backscattering, which allows the edge states to carry spin-polarized charges almost freely. Topological superconductors are simply TIs combined with superconductivity [15]. The superconductivity could be an inherent property of the TI system itself or brought to it by an adjacent regular superconductor through proximity effect. When the topologically protected edge states of a TI sits in a superconducting system, the particle-hole symmetry 8 of the excitations strictly pins one state to the Fermi energy. This special zero mode is also known as the famous Majorana fermion as it is its own Hermitian conjugate. Although the first experimental evidence for the existence of the elusive Majorana fermions was reported in 1D systems [118], and not in a 2D TI, the strong SOC and the subsequent band splitting still play a crucial role in those 1D systems. The semiconductor heterostructures, though they are the important ones, are not the only low-dimensional system exhibiting the interesting features associated with SOC and the related band and spin structure. Two-dimensional spin-orbit systems with a similar band and spin structure have been detected in the surface state of (typically heavy) metals and alloys, and have been synthetically realized in ultracold atomic lattices too. Here at the end of this section, we provide a brief outlook of what has been done in these two other categories of spin-orbit condensed matter systems. In 1996, LaShell et al. reported spin splitting of the surface state bands of Au(111) for the first time using Angle-Resolved Photoelectron Spectroscopy (ARPS) [102]. Later in 2001, high-resolution measurement of the (111) surface states of noble metals (Cu, Ag, and Au) [145] made the Rashba splitting on the surface of metals well known. Subsequently a spin-resolved photoemission measurement on Au(111) surface revealed the expected spin texture as well [85]. Since then, different kinds of measurement have been done on related systems where in some cases, very large SOC was reported such as in the surface alloy Bi/Ag(111) [7] or in the (111) surface of pure Ir [174]. It’s been also shown that the giant surface SOC of Ir(111) persists even if it’s covered with graphene layers, and its location with respect to the Fermi level is tunable by changing the number of graphene layers [154]. Along with Au(111) surface, W(110) has become the standard model for investigating Rashba SOC on the surface of metals [84, 149]. The W(110) also shows Dirac-like surface bands, which makes it particularly interesting for potential topological states that it can possess. It is proposed that the Dirac-like dispersion of the surface state of W(110) can be understood as a combination of SOC, strong charge transfer at the surface, and the relaxation in the first atomic layer [29]. Moreover, to study the spin-orbit split in the surface state of heavy metals such as Bi(111) and Sb(111), thin film coating of them on Si substrate has been used [167]. In the case of Bi, using an insulator substrate made it possible to investigate the effects of 9 Rashba-split surface states on its transport properties [168]. After this short review on the spin-orbit effects on the surface states of metals, now we turn to synthetic spin-orbit systems that can be designed in the ultracold atomic lattices. While in the actual material systems, SOC and other features of the system are hardly controllable and are mostly bound to their inherent properties of the underlying crystal structure, in ultracold atomic lattices, many parameters can be engineered in principle. In these synthetic lattices, not only the single particle Hamiltonian of the system can be artificially created, but also one can design how the individual atoms interact with each other. Since analyzing condensed matter interacting models (not even the actual material systems) is often near to impossibly complicated, these systems can also act as real-world quantum simulators for such models. The design flexibility that ultracold atomic lattices offer makes it possible to create a variety of SOC that cannot be found in the world of crystals [3, 90, 151]. They even allow for building systems that have no counterpart in any other field of physics. For instance, since spin-1/2 particles are strictly fermions, it would be hard to imagine a bosonic system with SOC. However, in ultracold atomic lattices, one can couple a pseudo-spin degree of freedom of some bosonic atoms to their motion to create an artificial bosonic system with SOC [6]. Realizing synthetic SOC in ultracold atom systems using laser-coupling was first implemented in 2009 in Spielman’s group [104] where it was developed further to create the peculiar spin-orbit-coupled Bose-Einstein condensate [105]. Generally, the temperature in the synthetic atomic lattice systems must be incredibly low (of the order of nano-Kelvin) such that the delicate closely spaced internal energy levels of the building blocks of the system can be singled out for further manipulation. The idea is to choose two internal states of each atom as the pseudo-spin state and couple it to a pair of counter-propagating laser beams. The laser beams’ wavelengths are picked in a way that they can induce a two-photon Raman transition in interaction with the atom and its pseudo-spin levels. But Raman scattering is, of course, inelastic and results in a small momentum transfer, which can be significant in that ultracold temperature. Then when the atom starts to move, the Doppler effect causes deviation in the perfect frequency match of the Raman transition. This in effect couples the atom’s momentum to its pseudo-spin and provides the potential 10 of engineering of all different kinds of SOC. In 2012, two collaborating groups from MIT and Cambridge created a fermionic cold atom lattice with an equivalent of a linear SOC [37]. They showed by direct spectroscopic measurement that the system has the very iconic band-split spectrum that is characteristic of 2DEGs with linear SOC. Campbell et al. [34] showed how by coupling three internal states of comprising atoms of the lattice, any combination of Rashba and Dresselhaus SOC can be achieved. Further development in principle can make the 3D equivalent of different forms of 2D SOC possible, many of which are technically impossible to realize in the real materials. Investigating interacting many-body systems of fermions is particularly interesting because of the dramatic effects of Pauli exclusion. For instance, we know that for a two-body fermionic problem in 3D, a threshold attraction is needed to form a bound state. However, according to the famous BCS theory in a many-body fermionic system, any infinitesimally small attraction is enough to create a sea of Cooper pairs. The smooth transition from a BEC of well-defined molecules to a truly many-body state of overlapping Cooper pairs in a standard BCS superconducting state is called BEC-BCS crossover [73, 184]. SOC can dramatically change both sides of such a crossover. Ultracold atom systems are promising tools to study such phenomena experimentally while invoking theoretical investigations [83]. An essential part of the present dissertation is to study this BEC-BCS crossover in the presence of the Rashba SOC (also known as non-Abelian gauge field in the coldatom community to differentiate it from the Abelian synthetic SOC that is technically less challenging to realize). Here in this work, the specific complication that was under our attention was to understand the effects of Coulomb repulsion in such a system, an effect that cold atom community is typically not concerned with as they usually deal with neutral fermionic atoms. With this note, next we move to the other half of our introduction that aims to create a background for the superconducting aspect of our problem. 1.2 Superconductivity The invention of the gas liquefier machine in 1895-6 independently by William Hampson and Carl von Linde opened the way for the vast field of low-temperature researches. In 11 1900, the Dutch physicist Heike Kamerlingh Onnes acquired a Linde machine for his laboratory, and it was in 1908 that his group at Leiden University successfully produced liquid helium at atmospheric pressure. Having liquid helium provided immediately makes it possible to build a cryogenic machine that can lower the temperature down to 4.2 Kelvin, the boiling temperature of helium at atmospheric pressure. Using the liquid helium cryogenic system, Onnes started to investigate the resistivity behavior of metals at very low temperature. It was in studying the low-temperature resistance behavior of mercury that they observed the shocking abrupt disappearance of mercury’s resistivity at 4.2 Kelvin, a historical discovery that he published in a paper in 1911 [127] and won the Physic’s Nobel prize for in 1913. Afterward, the same phenomenon that was later dubbed superconductivity was observed in many different materials. However, it took the scientific community quite a long while to develop a satisfactory theory for the phenomenon of superconductivity. In 1933, Walther Meissner and Robert Ochsenfeld discovered that superconductors repel magnetic fields [112], an observation that established the crucial connection of magnetism and superconductivity. Later in 1935, the London brothers explained that the Meissner effect could be understood by the tendency of the superconducting system to reduce the electromagnetic free energy [107]. The first successful theoretical attempt that captured the quantum mechanical nature of the phenomenon of superconductivity came from the phenomenological theory of Ginzburg and Landau that appeared in a paper published in 1950 [72]. Later using Ginzburg-Landau theory of superconductivity, Alexei Abrikosov elaborated on the two qualitatively different possible behaviors of superconductors in an external magnetic field that categorize superconductors into type I and type II superconductors [1]. In 2003, Abrikosov and Ginzburg won the Nobel prize in physics for their contribution in the theory of superconductivity. In the same year that Ginzburg-Landau published their theory, two experimentalist groups (E. Maxwell [110] and C. A. Reynolds et al. [147]) reported the isotope effect in the superconducting transition temperature of mercury. That implied that the nuclei sitting at the heart of crystal ions must play some role in the microscopic mechanism of superconductivity. Characteristic energy scale of lattice vibrations, quanta of which is called phonon, 12 is sensitive to ions’ mass in a very similar form to that which the isotope effect manifests in transition temperature. In fact, in 1950, Fröhlich had already suggested that electronphonon interaction is probably responsible for superconductivity [65]. In another paper in 1952 [66] (ignoring Coulomb interaction), he demonstrated that a canonical transformation can be used to reduce the electron-phonon interaction into an effective electron-electron interaction that is attractive when the interacting electrons via a phonon are in the same narrow range of energy. Later, his result was generalized by Bardeen and Pines to include the screened Coulomb interaction as well [11]. It was in this background that the brief paper of Leon Cooper came out in 1956 [41], which became the root of the term Cooper pairs. In that paper, he showed that if there is any effective attraction between a pair of electrons with opposite momenta sitting on top of a Fermi sea, then they form a bound state, implying that the ground state energy of such a system would be below the Fermi level by an amount proportional to the volume of the system. Following this work in 1957, finally, Bardeen, Cooper, and Schrieffer published their seminal microscopic theory of superconductivity in a comprehensive paper that addressed many aspects of the problem [10]. Their theory is one of the pinnacles of theoretical condensed matter to this day for which they won the Nobel prize in physics of 1972. The BCS result was then found by different people using different methods. Among the most important ones, Bogoliubov et al. showed that what BCS found from their variational wavefunction can be derived using field theoretical techniques [22]. In that work, they used a canonical transformation to obtain the elementary excitations of the superconducting state that is now known as Bogoliubov transformation. Being more general and systematic, the Bogoliubov approach was then used by others like de Gennes to study different forms of superconductivity [45]. A little later, Gorkov introduced a Green’s function approach that could derive both BCS result and the phenomenological Ginzburg-Landau theory [80]. Although before the BCS paper it was shown that even considering the Coulomb repulsion electron-phonon interaction can still produce a dominant effective attraction in a narrow range of energy, the interaction form in the original BCS model was a simple constant short-range attraction with an energy cutoff. The Gorkov’s Green’s function 13 method proved to be the most appropriate approach to address the dynamics and details of the effective interaction, especially the kind that was believed to have its roots in electron-phonon interaction. In the coming years, several physicists contributed to improving this aspect of the theory. For instance, Morel and Anderson showed how considering the Coulomb repulsion leads to the renormalization of the effective attraction by an amount that decreases as the ratio of Fermi energy to the phonon bandwidth increases [117]. Using Green’s function method, Eliashberg then considered retardation in electron-phonon interaction and developed a formalism that is now known under his name [59]. Later, Schrieffer et al. expanded on Eliashberg formulation to treat the Coulomb and electron-phonon interactions together and on the same footing [155]. Unconventional superconductivity is a term that is used to describe those superconducting states that do not fit under BCS theory. For instance, the systems in which the underlying pairing mechanism is something other than electron-phonon interaction form an important category of unconventional superconductivity. Studying of unconventional superconductivity has been a focus in condensed matter physics especially after the discovery of high Tc cuprate superconductors [14] as it is widely believed that some unconventional mechanism must be at work there. Since this work is partly concerned with the role of Coulomb interaction, among all proposed alternative pairing mechanisms, we would like to briefly mention Kohn-Luttinger mechanism, which quite counterintuitively solely stems from Coulomb interaction [97]. Before the short description of Kohn-Luttinger mechanism that follows, we should emphasize that although Coulomb repulsion is an essential part of our model, Kohn-Luttinger mechanism will not be present there. That is because, as we see, the Kohn-Luttinger mechanism relies on electron gas polarization, which is virtually absent in our model. Consider a free electron gas with a neutralizing background. Now if we bring a small perturbation in such a system (for example by introducing a test charge), the freely moving electrons will rearrange to minimize the total energy by screening the electric field generated by the small perturbative test charge. The screened Coulomb potential of the test charge will no longer decay as 1/r. In fact, the classical treatment of this problem will predict that the screened potential should decay exponentially. However, in 1952, Friedel showed that in a degenerate electron gas, the screened Coulomb potential would 14 have a very different tail that behaves as cos(2k F r + φ)/r3 [64], in which k F is the Fermi momentum and φ is some offset phase. The oscillatory part of this form, which is now known as Friedel oscillation, is crucial for the Kohn-Luttinger mechanism. Roughly speaking, the oscillations in the tail of Coulomb potential already imply attraction in certain distances. The crucial technical point is that the oscillations are rooted in the singularity of the Fourier transform of the polarization function at 2k F . Kohn and Luttinger separated this singular part of the Coulomb interaction from the smooth part and then studied their behaviors for different angular modes, “l”. It turns out that for large l’s, the singular part decays as 1/l 4 while the regular part decays exponentially. While the regular part is repulsive, the singular part can be attractive. In fact, they showed that for odd l’s, the singular part is always attractive while even for l’s, it is attractive if √ U (0)/U (2k F ) > 3 − 1, where U (q) is the regular part of the Fourier transform of the screened Coulomb potential [97]. They also applied their theory to the case of 3 He, which was then believed to have a pairing channel at l = 2, and found incredibly small Tc of the order of 10−17 Kelvin. However, a couple of years after the work of Kohn and Luttinger, in 1968, Fay and Layzer found a more reasonable Tc (of the order of milliKelvin) assuming l = 1 [62], which was also closer to the experimental result that came a few years after that [128]. The experiment also confirmed that the pairing occurs at l = 1 [128]. To close our introduction to the role of Coulomb interaction in superconductivity, let us point out that although the original work of Kohn and Luttinger was considering a threedimensional electron gas with a quadratic dispersion, a similar scenario can still happen in two-dimensional electron gas (2DEG) [39] and indeed in the case of a 2DEG with Rashba SOC [101]. Speaking of superconductivity in two-dimensional systems, now we would like to provide a short review on this topic as well, especially since the specific model that we studied in this dissertation is a two-dimensional system. Although since the beginning of the modern studies on superconductivity in the late ’50s and the ’60s, 2D superconductors were deemed to be in a very fragile state of matter (if even possible), theoretical works considered such systems as early as 1964 [71]. Ginzburg and Kirzhnits speculated the possibility of superconducting state forming in the surface state of a solid. They even proposed two possible experimental settings to realize such a state, one on the surface of a dielectric and one on a metal [70]. 15 Before 1966, the weakness of Coulomb screening in 2D plus the detrimental effects of impurities in two-dimensional superconductors were the key reasons for physicists to be skeptical about 2D superconductivity. In 1966, Mermin and Wagner published their famous work that stated that in low-dimensional systems, it’s impossible to have a longrange order if it requires breaking of a continuous symmetry [113]. And according to GL theory, superconducting order is such an ordered phase. However, the Mermin-Wagner skepticism was later alleviated by works of Berezinski [18] and Kosterlitz and Thouless (BKT) [100]. They showed that lack of correlation in 2D could be associated with the creation of vortex-like defects, which come in attracting pairs of vortex-antivortex. Below a certain temperature (that is lower than what a mean-field theory would predict for a 2D superconductor), these pairs are bound, which effectively still allows for a long-range ordering [100]. Despite the fact that, as it’s pointed out by several researchers [30, 170], observing BKT transition in realistic material systems is very elusive, it has been observed in the superconducting interface of LaAlO3 and SrTiO3 at the very low temperature of 188 mK [148]. Although according to BKT, 2D-superconductivity is possible in principle, it is still highly sensitive to disorder. In fact, in 2D superconducting systems, sufficient amount of disorder can lead to a superconductor-insulator transition [75]. This requires extremely clean and disorder-free systems to realize 2D superconductivity, which has been the major technical challenge. The recent technological development has made it feasible to create clean two-dimensional superconducting systems. In particular, molecular beam epitaxy (MBE) method in fabricating materials and ultrahigh vacuum (UVH) and low-temperature measurement systems have offered a significant advancement. For instance, now it’s possible to deposit a few perfect metallic layers (such as Pb) on top of a silicon substrate with extraordinary control on the number of layers. It’s been observed that such two-dimensional systems can show superconductivity even with a single atomic layer [129, 185]. Another point of interest in 2D superconductors comes from the high Tc cuprate superconductors where the Coulomb interaction is also believed to play an essential role. Bollinger and Bozovic successfully observed superconductivity in a single-unit-cell layer of the famous cuprate La2− x Srx CuO4 , which is an evidence for the two-dimensional na- 16 ture of superconductivity in the high Tc cuprates. They also suggest that the transition temperature in a single layer of CuO2 can be as high as the transition temperature in the bulk [25]. Also, since 2D superconductivity in many insulators and semiconductors can be very sensitive to the level of carrier doping, the new techniques of gate-induced carrier doping have made it possible to create field-induced 2D superconductivity in materials such as ZrNCl and SrTiO3 [171, 183]. Speaking of SrTiO3 , we should mention that this insulating material was used as the substrate to materialize one of the most exciting results in the field of 2D superconductivity. MBE technique was used to grow FeS superconductor on SrTiO3 layer by layer. Implementing such a setup, Wang et al. observed that the FeS transition temperature increases significantly from the bulk value of 8 Kelvin up to 100 Kelvin as the number of layers decreases down to one single layer [178]. It is believed that in this system, the substrate plays a vital role in the Tc enhancement, and several different mechanisms have been proposed for the possible nature of this effect [106, 179]. We also shouldn’t forget about one of the most important categories of 2D materials, namely the atomic sheets. As it was first shown for the case of graphene, exfoliation technique can be used to isolate a single atomic sheet of a layered material [123]. Different groups are currently studying 2D superconductivity in atomic sheet systems such as graphene [109] and the single layer of transition-metal dichalcogenides such as NbSe2 [172] and MoS2 [42]. Another category of 2D materials under investigation seeking for their superconducting behavior are those involving organic conductors [115]. As Baskaran points out searching for more novel 2D superconductors, we shouldn’t limit ourselves to the well-known 2D materials, as unconventional pairing symmetries can be realized in many different systems [12]. Proposing a possible novel mechanism, he argues that a Wigner crystal state in a low-density system with strong Coulomb repulsion can melt down to form a superconductor state as a result of additional doping and quantum fluctuations. Studying the effects of broken symmetries in 2D superconducting systems is another direction of investigation that is also related to the work of this present dissertation. For instance, as we mentioned earlier in this introduction, breaking of the inversion symmetry in 2D systems follows by Rashba SOC that can affect pairing symmetry [79]. Or as it’s also 17 suggested, the presence of Rashba SOC can lead to the evolution of a conventional s-wave pairing to the hybrid state of triplet-singlet pairing [152]. Moreover, as our present work suggests, Rashba SOC in a 2D low-density system can also bring about the phenomenon of BES-BCS crossover, a phenomenon that is the subject of the last bit of this introduction. BCS in their seminal paper emphasized the strong overlap of Cooper pairs in their theory. In particular, they differentiated it from Schafroth-Butler-Blatt theory of superconductivity, which relies on Bose-Einstein condensation of nonoverlapping composite bosons [157]. Qualitatively, the two regimes can be understood by comparing the pair size with the average interparticle distance. In the BCS regime, the pair size is much larger than the interparticle distance, which leads to strongly overlapping pairs. On the other hand, in the BEC limit, the pairs are smaller than the average interparticle distance, and at sufficiently low temperature, they can undergo Bose-Einstein condensation. Figure 1.3 is demonstrating the limits of this crossover schematically. Naturally, there are two ways to go through such a crossover. First, one can imagine a system with a fixed interaction, as is the case in our model, while the particle density is changing, for example by tuning the chemical potential. Second, as it is realized in the ultracold atomic gases, one can imagine a tunable interaction in a system with fixed particle density. In the latter, the crucial tunable parameter of the interaction is usually the scattering length that can have a negative or positive value. In fact, in experimental settings, it is the inverse of scattering length that can be smoothly varied. The resonance point at which the scattering length diverges and changes sign is known as the unitary regime and is believed to exhibit universal behavior. The unitary point is often considered as the representative of the intermediate regime in the crossover. The fact that there is no true phase transition between the BCS and BEC limit is theoretically nontrivial, and in fact, it’s believed that for certain pair symmetries, there should be a phase transition between the two regimes [103]. However, qualitatively arguing as long as the class of broken symmetry is the same on the two sides of the spectrum, there should be no phase transition in between, and the same wavefunction with smoothly varying parameters should be sufficient to describe the system across the crossover. After general acceptance of the successful superconductivity theory of BCS, it took a while until a few attempts were made to see the two sides of what was later called 18 Figure 1.3. Schematic view of BCS-BEC crossover. In the BCS limit, pairs are strongly overlapping while in the BEC limit, pairs are molecular-like and their size is smaller than the interparticle distance. In the intermediate state, the pair size is comparable with the interparticle distance. BCS-BEC crossover as extreme limits of a unifying theory. In 1965, studying the possible condensation of excitons in semiconductors, Keldysh and Kopaev considered such a crossover [94]. A year later, with purely theoretical motivation, Popov studied the different limits of a Bose gas produced by bound states of Fermi particles [132]. A couple of years after that, in 1969, Eagles considered such a possibility in the context of superconductivity in thin-film semiconductors [57]. The seminal work on the topic came out in 1980 by Legget where he systematically studied BCS-BEC crossover in a gas of attractive interacting diatomic molecules [103]. Legget used a BCS treatment at zero temperature assuming that the scattering length for the interaction is tunable. A few years later in 1985, Nozières and Schmitt-Rink worked out the problem at finite temperature and above the transition temperature [124]. The problem received more attention after it was perceived that in the newly discovered high Tc cuprate, the pair-size could be comparable with the interparticle distances [52, 82, 135], and the two-dimensional case was also considered from the beginning of the new wave of interest [134]. Since such a feature in principle can be present in any fermionic system with attractive interaction, soon the topic also emerged in the field of nuclear physics and in particular in the context of deuteron formation [8]. Although the study of BCS-BEC crossover has its roots in a variety of systems, it has 19 only been successfully realized in the ultracold atomic systems. In 1995, the quantum state of Bose-Einstein condensation was realized for the first time in a couple of systems of atomic gas of alkali metals (87 Rb [4], 7 Li [28], and 23 Na [44]). Due to technical difficulties in achieving ultracold temperature for fermionic atomic systems, it took a while until the first fermionic degenerate gas was created for 40 K in 1999 [46] and then for 6 Li in 2001 [158, 169]. Consequently, a couple of years later, finally for the first time, BCS-BEC crossover was realized by several different groups in different fermionic atomic gases (for 40 K [144] and 6 Li [27, 186]). The crucial tool to control the interaction in the ultracold atomic systems is a Feshbach resonance, which enables one to change the magnitude and the sign of scattering length by varying an external magnetic field. Such a controllable interaction was necessary and had been implemented in fermionic cold atomic gas prior to the first experimental realization of BCS-BEC crossover [26, 125]. Finally, another direction of research that has been made possible by the advent of ultracold atomic gases comes from combining them with optical lattices, which had already been developed in the ’90s [88, 177]. Filling an optical lattice at ultracold temperature with fermionic atoms creates a highly tunable fermionic lattice, which can be used to simulate strongly correlated electronic systems. In particular, as we mentioned earlier, recently several different groups were able to simulate the equivalent of different classes of SOC on ultracold fermionic optical lattices [37, 104, 105]. For instance, although to the best our knowledge, it hasn’t been experimentally implemented yet, it was proposed that the BCS-BEC crossover can be derived in an ultracold fermionic lattice with the equivalent of Rashba SOC, without changing the interaction or density and solely by increasing the strength of SOC [83]. Now that we prepared a brief historical and conceptual context on the spin-orbit coupling in condensed matter systems and the phenomenon of BCS-BEC crossover, next we are going to develop a specific model to study its properties. It’s going to be a 2DEG with Rashba SOC induced by broken inversion symmetry. We are going to assume a fixed constant attractive interaction in the system that will be the origin of a possible superconducting state. We will show that a BCS-BEC crossover occurs in our model by tuning the chemical potential. Moreover, the inclusion of an externally screened Coulomb interaction provides another way to go between different phases of the model and across the BCS-BEC 20 crossover. Having a 2D model, we will also investigate the effects of fluctuations in the model and discuss the BKT transition. CHAPTER 2 DEFINING AND FORMULATING THE PROBLEM Now it’s time to construct the specific problem that we want to address and make a mathematical model for it. Imagine that there is a two-dimensional electron gas (2DEG) possibly realized by some 2D material or a heterostructure, a system in which the inversion symmetry is broken and therefore some spin-orbit coupling (SOC) is present. We take the SOC to be of the simple form of Rashba type. As another simplifying assumption, we take the constant of SOC to be constant. The other element of our simple system will be a conducting gate on top of the 2DEG. The gate is going to serve two purposes, first as an external screening mechanism and second as a tool for controlling (electro-)chemical potential. A schematic of the setup is sketched in Figure 2.1. In fact, interestingly enough, it is shown that for large enough SOC such a configuration can itself create a bound state in the system and induce an inherent pairing mechanism [69, 108]. However, to add the other ingredient of our model, namely the attraction mechanism, we do not explicitly rely on that elegant induced pairing mechanism. Instead, we simply postulate an attractive interaction. To make it even simpler, we assume the (mathematically) simplest form of attraction, namely the contact attraction. These two, SOC and the attraction, are the main features of our model, a model to study insulator-superconductor transition in a 2DEG with Rashba SOC. As the last ingredient, we take into account the electrostatic Coulomb repulsion, which, as we will see, is crucial to forming a minimum realistic model. It will also produce the main mathematical challenge of the problem. In fact, our main contribution to the literature is to study the effects of the electrostatic repulsion on the transition, an effect that is not negligible in the dilute limit that we are going to consider. 22 Figure 2.1. Schematic for the gated 2D layer system. 2.1 Single particle Consider a system of free electron gas with effective mass meff = 1/2κ that is expressed in terms of κ only for convenience. The regular kinetic energy in terms of momentum p would be κ p2 that is the same for both spins. If the inversion symmetry in z-direction is broken, its Hamiltonian will also have a spin-orbit coupling term. We consider a simple Rashba SOC that famously has the form α(~p ×~σ).ẑ [138] in which spin-dependence manifests itself in the vector of Pauli matrices ~σ. Moreover, for us, α, the SOC parameter that has the dimension of speed, is a constant that is determined by the underlying system; the larger the α, the stronger the spin-orbit effect. Therefore, the Hamiltonian that describes single particle behavior of the system in momentum space is H0 (~p) = κ p2 + α( p x σy − py σx ), (2.1) which is not a diagonal 2 × 2 matrix. In order to find the energy spectrum and the spin pattern of this Hamiltonian, one has to diagonalize it by finding its eigen-energies and eigen-vectors. It’s straightforward to find that the two eigen-energies are given by ε ± = κ p2 ± αp = κ ( p ± p0 )2 − ER . (2.2) where p0 = α/2κ = αmeff is Rashba momentum and ER = κ p20 is Rashba energy. These two bands of H0 are shown in Figure 2.2. The corresponding eigen-vectors of ε − and ε + are respectively given by 1 χ− = √ [ie−iθp , 1] 2 1 χ+ = √ [1, ieiθp ] 2 (2.3) 23 Figure 2.2. Main: Side-view of the two 2D Rashba spin-orbit split bands for Λ > 0 situation. The insert shows the top view of the ring-like Fermi surface for Λ + Φ < 0 case; see Eq. (4.22) and discussion related to it. where eiθp = pˆx + i pˆy and so θp is the angle momentum p makes with x̂-axis. Also, one can use Eq. (2.3) to see the spin pattern of each of the bands. Doing so, we see that they are both in plane while χ− is in −θ̂ direction and χ+ is in +θ̂ direction, where θ̂ is the standard angular direction of polar coordinate. The lower band, ε − , which resembles a moat, has a minimum at p0 when ε − ( p0 ) = − ER . Therefore, the energy of a single particle in the moat-shaped band, measured from a given chemical potential, can be written as ξ p = κ ( p − p0 )2 + Λ, (2.4) where Λ is the minimum energy of the moat band measured from the chemical potential (see Figure 2.2). This completes all the important information that we need from the single particle spectrum of the model. Now we can move on and add the interactions to the play. 2.2 Projected interactions To start building the interaction terms, let’s start by considering the general form of density-density interaction 24 0 Hint = 1 2V Ucp† +q,σ cp† 0 −q,σ0 cp0 ,σ0 cp,σ , ∑ ∑ 0 0 (2.5) σσ ∈↑↓ pp q where U is a general interaction form in k-space and cp,σ is the annihilation operator of a particle with momentum p and spin σ. We are interested in a low-density situation when the chemical potential lies below e+ band and therefore, the upper band doesn’t play any significant role. Therefore, we are going to exclude the upper band and only consider contributions made by the lower band. In other words, we only keep the projection of (2.5) on the lower band. In order to do that, we have to use the inverse form of (2.3): c↑,p = c↓,p = i 1 h √ −ieiθp c−,p + c+,p , 2 i 1 h √ c−,p − ie−iθp c+,p , 2 (2.6) to rewrite (2.5) in terms of c− and c+ and then only keep terms with c− . Since now we only have one band, we are going to suppress the band index ”−” and simply use c− ≡ c. After careful algebra, the projected interaction Hamiltonian is found to be Hint = 1 8V ∑0 Ucp† +q cp† 0 −q cp0 cp (1 + e−iθ 0 +iθ 0 − p p q )(1 + e−iθp +iθp+q ). (2.7) pp q This is now a single band Hamiltonian that includes an extra prefactor in front of U. Next we are going to identify two different channels in this generic form, one attractive and one repulsive. 2.2.1 Attractive Cooper channel We choose the Cooper channel to be the provider of the attractive part of the interaction. By Cooper channel we mean the standard channel in which the interacting electrons are sitting at the opposite momenta p0 = −p [41]. Note that spins of these electrons are also opposite to each other so we will have no trouble with postulating a contact attraction. In this channel, the phase factor in (2.7) reduces to (1 + e−iθp +iθp+q )2 . However, since cp c−p = −c−p cp , even in momentum terms vanish and only terms with the phase factor 2e−iθp +iθp+q survive. Also for the sake of simplicity, rename p + q ≡ p0 . Next, we assume that the interaction is a contact attraction, which means it can be represented by a constant 25 in Fourier space. Taking the constant, which is the strength of the attraction, to be −2g, we can write the attractive interaction that is carved out of (2.7) as Hp = − g 2V ∑0 e−iθ +iθ 0 cp† 0 c†−p0 c−p cp . p p (2.8) pp This will be the part of interaction that creates superconductivity. 2.2.2 Long-range Coulomb channel For the Coulomb channel, however, we work with the full interaction form presented in (2.7). The significance of the phase factor, as we show in the rest of the paper, shall depend on the external screening mechanism that we introduce below. Therefore, the repulsive part of the Hamiltonian maintains the general form in (2.7). Hc = 1 8V ∑0 Ueff (p, p0 , q)cp† +q cp† 0 −q cp0 cp , (2.9) pp q where Ueff = Uq (1 + e−iθp0 +iθp0 −q )(1 + e−iθp +iθp+q ), (2.10) in which Uq is the Fourier transform of the (possibly screened) Coulomb repulsion and the phase factor comes from projecting the interaction on the “moat band”. The long-range nature of the Coulomb interaction brings up the issue of screening in a system of charged Fermions. In the two-dimensional (2D) system with low-electron density, screening of Coulomb interaction via polarization is not effective. Instead, we assume that our 2D layer of Rashba material is screened by a proximate conducting gate, see Figure 2.1, which also provides a simple way to control the (electro-)chemical potential of the system by varying the gate voltage. The effect of the conducting gate can be understood by using the image charge technique. Image theorem tells us that any local charge introduced in the layer creates an imaginary charge with the opposite sign at the position of its mirror picture on the other side of the gate. That means the local charge and its image form a dipole of size d. Therefore, in the presence of the conducting gate, the effective electrostatic interaction becomes like dipole-dipole interaction. Dipole-dipole interaction has a shorter range than bare Coulomb as it decays like 1/r3 rather than 1/r, where r is the distance between two local charges. e2 U (r ) = 4πe 1 1 −√ 2 r r + d2 , (2.11) 26 where e is dielectric constant of the underlying material and its surrounding, and d/2 is separation between the gate and the layer. The Fourier transform of U (r ) (that is a two-dimensional integral) can be found analytically and it reads as: Uq = e2 (1 − e−qd ). 2eq (2.12) Observe how the extra factor of 1 − e−qd removes the singularity of Fourier transform of bare Coulomb in 2D at q = 0. This is another manifestation of Coulomb becoming short-range, as small q corresponds to large r. We have completed building every important element of our model. The full Hamiltonian H = H0 + H p + Hc is a combination of (2.1), (2.8), and (2.9). And the Uq in (2.9) is given by (2.12). Now that all the mathematical details of the problem are exactly specified, we move on to try solving it. The three subsequent chapters will present three different approaches to solve this model. CHAPTER 3 SINGLE PAIR WAVEFUNCTION In order to study the Hamiltonian we built in Chapter 2, we start by a simple model in which there is only a single Cooper pair living in the system. This is particularly plausible when the noninteracting system is just a vacuum; the chemical potential is below band edge (Λ > 0) and therefore, without the interaction, the system is an insulator (see Figure 2.2). In this situation, we can ask what would a pair of electrons do if they are introduced to this system. For this reason, in this chapter, we are going to consider a generic wavefunction that describes a single Cooper pair, and we assume the noninteracting system to be an insulator; that is to take Λ > 0. Recall that we have two controlling parameters in our model, both of which can be tuned through the conducting gate; the gate distance, d, controls screening and hence the strength of Coulomb repulsion and also we assume that Λ can be tuned by applying tunable voltage to the gate (see Figure 2.1). It is clear that since there is finite binding energy required for a pair to enter the system, the spatial extent of such a pair is also finite, being determined by roughly the inverse binding energy. On the other hand, for the range of parameters where the pairs are just (energetically) able to enter the system, their density is vanishingly small, and one indeed deals with the case of a dilute Bose gas of pairs. The diluteness of the pair gas implies that to find the conditions at which the BEC occurs, one only needs to determine the conditions required for the first pair to enter the system. Therefore, in this section, we solve the single-pair Schrödinger equation to determine when the pair bound state goes under the chemical potential for pairs. Since the chemical potential plays only an auxiliary role in such considerations, setting the value below which the bound state energy needs to go, one can formally solve the single-pair problem even for Λ < 0. This is especially relevant in the case with Coulomb repulsion, which effectively increases the pair fugacity above 2Λ, hence Λ = 0 is not special within 28 the single-pair problem in that case. However, the physical starting point – a single pair in the system – is not correct in the Λ < 0 case, since one must start with a Fermi sea. Hence, we do not consider it. The appropriate treatment is given in Section 4. 3.1 Without Coulomb repulsion Let’s start our study from the simple case in which the Coulomb part is not present. Consider a situation when the chemical potential is modeled by an external particle reservoir. And as we mentioned above, we assume it is below the band (Λ > 0). That means in the absence of any attraction, particles prefer to stay in the reservoir. However, if we introduce the Cooper pair attraction channel into the system, there will be a bound state for the pairs and then for a strong enough attraction, that bound state might go below the chemical potential. Therefore, the electrons would prefer to pair up and populate that bound state as so-called pre-formed pairs. This is basically the mechanism in the Bose-Einstein-Condensate (BEC) limit where you can think of the electronic state as a condensate of a bunch of well-defined pairs. The wavefunction we use to describe a single Cooper pair is going to be a superposition of Cooper pairs at different possible momenta. Coefficients of these pairs are the degrees of freedom of the problem and hence defining the subset of the Hilbert space that we selected. Mathematically it can be written as |Ψi = ∑ ap cp† c†−p |0i, (3.1) p in which cp† creates an electron at the momentum p in the moat band and a set of coefficients ap determines the wavefunction. The form of this wavefunction is dictated by the form of the pairing Hamiltonian, Eq. (2.8), and in the absence of the Coulomb repulsion (when H = H0 + H p ), the wavefunction (3.1) is an exact eigenstate of the total Hamiltonian. The Schrödinger equation H |Ψi = E|Ψi gives us the following form for the bound state wavefunction in the momentum-space αp = g 1 2ξ p − E V c ∑ αk = 2κ ( p − p0 )2 + 2Λ − E , (3.2) k where αp = ap e−iθp and c is a constant given by the integral over k. Solving Eq. (3.2) self-consistently leads to a simple integral equation for E as follows: 29 1= g V 1 ∑ 2ξ p − E , (3.3) p in which ξ p = κ ( p − p0 )2 + Λ is our moat band single particle energy spectrum. Approximating the integral is straightforward and gives us E = 2Λ − g2 p20 . 8κ (3.4) A negative value of E implies that it is energetically favorable for a pair to enter the system. Therefore, the condition E = 0 determines the critical value of the coupling constant, g, at which condensate forms in the system (at a given value of Λ). It yields the critical interaction strength of gc0 √ 4 κΛ = . p0 (3.5) This results coincides with that of Ref. [76], which was obtained using field-theoretical methods. Here, we just emphasize the very simple nature of this problem, which reduces to calculating a pair bound state. At the critical point, when the bound state emerges, the corresponding wavefunction has finite spatial extension – the pair size. To get a scale estimate of this length scale, first −1 we realize from (3.2) that at the critical point, α p ∝ κ ( p − p0 )2 + Λ , which is a peak in √ momentum space with the width of Λ/κ. However, since the real space is the conjugate space of momentum space, one can expect that the pair size in the real space follows the inverse scale that is r ζ= κ . Λ (3.6) Following the same logic, one can tell by increasing g and deeper ground state on the negative side, the pair size shrinks. This would complete our analysis of the simple case of having no Coulomb repulsion. Next, we study the effects of Coulomb repulsion on the behavior of the pair wavefunction. 3.2 Effects of Coulomb repulsion Now if we include the Coulomb repulsion, Hc , in the Hamiltonian, the Schrödinger equation becomes (2ξ q − E)αq + where Ũq,k = Uq−k (1 + cos(θq − θk ))/2. 1 V g ∑ Ũq,k αk = V ∑ α p , k p (3.7) 30 The key difference brought about by the Coulomb repulsion is the fact that now the bound state formation requires a finite strength of the attraction. In the case of pure attraction, the bound state exists at any positive value of g, see (3.3), while a finite critical value gc0 (3.5) follows from the condition that the bound state must be 2Λ deep for pairs to enter the system. Below, we will refer to the critical value of g needed for the bound state formation as gcb , keeping the notation gc for the value of g that marks the onset of superconductivity. The form of solutions to this equation depends on the strength of screening. One should distinguish the limits of strong, dp0 1, and weak, dp0 1, screening, which we discuss in detail below. 3.2.1 Strong screening When the screening is strong, i.e., when dp0 1, Uq (defined in (2.12)) can be approximated by the constant e2 d/2e, reflecting the fact that it becomes a short-range contact-like interaction. Therefore, in that case, the Coulomb term in (3.7) can be absorbed in the constant attraction term on the right-hand side. Absorbing the Coulomb term, one should note that due to the presence of the angular factor, (1 + cos(θq − θk ))/2, it would diminish by a factor of 1/2 over the angular integral. With this consideration, we obtain a renormalized effective attraction geff = g − e2 d . 4e (3.8) It is obvious that the value of g required for the formation of a bound state is gcb = e2 d . 4e (3.9) In turn, just like in the case of pure attraction, the onset of superconductivity is marked by the value of g required for the bound state to cross the value of the chemical potential for pairs, 2Λ. As a result, strong screening of the Coulomb interaction by the conducting gate results in a weak enhancement of the critical strength of attraction gc , gc = gc0 + Ec d , 2p0 (3.10) where gc0 is given by Eq. (3.5), and we defined Ec = e2 p0 /2e as the characteristic Coulomb energy scale in the moat band. 31 In Figure 3.1, the analytic approximation (3.10) is compared with the results found by solving the Schrödinger equation (3.7) numerically. One can see that the strong-screening results – the linear in d enhancement of the critical attraction strength – persists to dp0 ∼ 0.1. 3.2.2 Weak screening In the limit of weak screening , dp0 1, the Coulomb repulsion of Eq. (2.12) is strong and q-dependent. In order to acquire an analytical understanding in this case, we use two approximation schemes. First we take a variational approach. Using |Ψi defined in (3.1) and the full Hamiltonian (H = H0 + H p + Hc ), the variational energy can be written as Evar 2 ∑p α2p ξ p + hΨ| H |Ψi = = hΨ|Ψi 1 V ∑p,q αp αq (Ũp,q − g) . ∑p α2p (3.11) Inspired by the bound state wavefunction obtained in the case without the Coulomb interaction, Eq. (3.2), we propose the following variational form for the bound state wavefunction in momentum space: αp = c , ( p − p 0 )2 + ζ −2 (3.12) in which c is the normalization constant, and ζ is the variational parameter that has the meaning of the pair size in real space. In the parameter regime of interest, when p0 ζ 1, the wavefunction (3.12) is peaked around p0 , the peak width being ζ −1 p0 . In this limit, approximating every integral in Eq. (3.11) is straightforward, except for the Coulomb one, which requires some care. For the Coulomb integral, after angular integration, we find 1 V VEc ∑ αp αq Ũp,q = (2π )2 p0 p,q Z pmax 0 dp Z pmax 0 dqpqα p αq K( p, q), (3.13) where the Coulomb kernel K( p, q) is defined as 1 K( p, q) = 2π Z 2π cos2 (θ/2)(1 − e−d| p−q| ) 0 | p − q| dθ, (3.14) θ being the angle between p and q. In order to proceed with the integral (3.13), below we construct an approximation of the Coulomb kernel K( p, q) defined above in (3.14). First on should notice that the function K is the largest when p − q ≈ 0; that is around the diagonal of the p-q plane. Moreover, all the integrals are important around p0 . Therefore, we can equivalently say that we are interested in K( p) ≡ K( p0 + p/2, p0 − p/2). Second with p p0 , we can neglect the cos2 (θ pq /2) term in the definition of K(q, k ). 32 6 ×10−1 4 numerical result for Λ = 0 (gc − gc0 )/κ 5 analytical approximation numerical result for Λ = 0.1ER numerical result for Λ = 0.05ER 3 2 1 0 0.0 0.2 0.4 dp0 0.6 0.8 1.0 ×10−1 Figure 3.1. Comparing numerical and analytic d-dependence of Coulomb correction to gc (Eq. (3.10)) for Ec = 10ER and for three different Λ’s in the range dp0 1. We now calculate K( p) for several important values of its argument, starting with the value of K(0). To this end, we set p, q → p0 in Eq. (3.14), and introduce a new variable t = sin(θ/2), such that dθ ≈ 2dt, to obtain 1 K(0) ≈ π p0 Z 1 0 dt 1 − e−2dp0 t 1 ≈ ln(4β2 d2 p20 ), t 2π p0 (3.15) where β = eγ with γ ≈ 0.58 being the Euler-Mascheroni constant, and to write the second approximate equality, we neglected O(e−2dp0 ) terms, which are negligible in the weak screening limit, dp0 1. We now turn to the behavior of K( p) for 1/d p p0 . In this limit, the exponential term in the definition of K( p, q), Eq. (3.14), can be neglected, and we arrive at K( p) ≈ 2 π Z 1 0 dt q 1 4p20 t2 + p2 ≈ The accuracy of the final expression here is O( p2 /p20 ). 16p2 1 ln 2 0 . 2π p0 p (3.16) 33 Combining the two limits, Eqs. (3.15) and (3.16), we arrive at the following interpolating function that connects them, and serves as a good approximation to the full Coulomb kernel. K(q) ≈ 4β2 d2 p20 1 , ln 2π p0 1 + β2 d2 q2 4 (3.17) Using (3.17), the integral in (3.13) can be approximated by extending the p and q integrals around p0 and using pdp ∼ p0 dp to get 1 V ∑ αp αq Ũp,q p,q Z ∞ Z ∞ ln(1 + β2 d2 ζ −2 ( x1 − x2 )2 )/4 c2 VEc ζ 2 1 . dx2 ∼ dx1 ln 2βdp0 − 4π 2π 2 −∞ ( x12 + 1)( x22 + 1) −∞ (3.18) In order to carry out the integral in (3.18), we use the following identity ln(1 + ay) = Z a 0 ds , s + y −1 (3.19) to rewrite the integral as 1 2π 2 Z β2 d2 ζ −2 /4 ds 0 Z ∞ Z ∞ −∞ −∞ dx1 dx2 ( x1 − x2 )2 = ( x12 + 1)( x22 + 1)(1 + s( x1 − x2 )2 ) Z βd/2ζ 0 βd 2du = ln(1 + ), 1 + 2u ζ (3.20) where the integrals over x1 and x2 are done by using residue theorem, and then we used √ u = s to obtain the final result. Finally, using the result in (3.20), the variational energy can be written as Evar = 2Λ + 2κ gp0 Ec 2βdp0 − + ln . ζ2 ζ π p0 ζ 1 + βd/ζ (3.21) Minimizing the energy, we find that the optimal pair size, ζ ∗ , can be found from the following equation: 4ER Ec = gp20 + p0 ζ ∗ π βd 1 1 + ln + . βd + ζ ∗ 2p0 ζ ∗ 2βdp0 (3.22) One can think of (3.22) as an equation for 1/p0 ζ ∗ in which by increasing g, a solution appears at a certain attraction that we call gcb . By analyzing behavior of Eq. (3.22) at 1/p0 ζ ∗ → 0, one can see that for d larger than certain dc , the solution appears with a 34 finite size: a rather peculiar behavior. With a simple algebra, it can be seen that such dc is given by d c p0 = 2πER . βEc (3.23) Figure 3.2 demonstrates this peculiar behavior of the Schrödinger equation (3.7) through its numerical solution. One can solve (3.7) as an eigenvalue equation and look for the eigenvector α p with the lowest energy. In the absence of a bound state, such α p is just a delta function representing a pair of free particles in the real space. At gcb bound state forms, however, it can be seen that such a bound state emerges with a finite size instead of coming from infinity. Assuming the case that the solution exists, using a simple algebra, one can find that the energy of the corresponding bound state is given by Eb = 2Λ − Ec βd 2κ + . ∗ 2 ∗ ζ π p0 ζ βd + ζ ∗ (3.24) However, to form a BEC of pairs, two conditions have to be satisfied. First, a bound state should exist that is equivalent to having a solution for (3.22). Second, the corresponding energy of the bound state has to be lower than the chemical potential. The two conditions together imply that to find the critical attraction, gc , at which superconductivity kicks in, one has to solve (3.22) and Eb = 0 together. Following that logic, one has to first find the critical pair size by solving: 2Λ = 2κ Ec βd − , 2 ζc π p0 ζ c βd + ζ c and then from (3.22) and in terms of ζ c , we find the critical gc as 4κ e2 βd 1 1 gc = − + ln + . p0 ζ c 2πep0 βd + ζ c 2p0 ζ c 2βdp0 In particular, for the special case of Λ = 0, the explicit expression for gc reads e2 4ER 2κ gc = 1 + ln − . 2πep0 Ec βdp0 (3.25) (3.26) (3.27) Here one should note that to stay consistent with our assumption p0 ζ 1, we need to require Ec < ER , which could be achievable if p0 and e are large enough. Figure 3.3 compares the analytical approximation given by (3.26) in the limit of d → ∞ and the result found by solving the Schrödinger equation (3.7) numerically; the analytical 35 0.8 g/κ = 0.1724 αp 0.6 0.4 0.2 0.0 −1.00 −0.75 −0.50 −0.25 6 0.00 0.25 0.50 ×10−2 0.75 1.00 ×10−2 g/κ = 0.1725 αp 4 2 0 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 (p − p0 )/p0 0.75 1.00 −2 ×10 Figure 3.2. The numerical solution of the Schrödinger equation (3.7) for Ec = ER /10 and dp0 = 1000 and two incrementally successive values of g. The numerical value of g/κ is printed in each panel. It can be seen how the bound state emerges with a finite size in the lower panel. result of the neutral case (Eq. (3.5)) is also included as a point of reference. It is seen that, although it’s not a perfect match, the analytical variational approach captures the qualitative behavior of the critical gc in the presence of strong not-screened Coulomb repulsion. As the second approach to study the solutions of Eq. (3.7), first we realize that a bound state solution of that equation must have s-wave symmetry; that is to say α p doesn’t have θ-dependence. Noting that we can integrate the angle out and obtain 36 2.00 1.75 1.50 gc /κ 1.25 1.00 0.75 analytical approximation numerical result without Coulomb repulsion 0.50 0.25 0.00 0.0 0.2 0.4 0.6 0.8 Λ/ER 1.0 ×10−1 Figure 3.3. Comparing numerical and analytical (Eq. (3.26)) Λ-dependence of gc in no-screening case and for Ec = ER . The dashed curve shows the result in the absence of Coulomb repulsion (Eq. (3.5)). 2ξ q αq − gp0 Z p α p + Ec Z k K(q, k)αk = Eαq , (3.28) where the Coulomb kernel K is defined in (3.14), and we introduced a shorthand notation Z p ≡ Z ∞ dp −∞ 2π . (3.29) Furthermore, as explained above, we are interested in finding two-electron bound states with vanishing energy, E → 0, to find the boundary of the normal-superconducting transition. For Λ ER , only momenta in the close vicinity of p0 participate in the formation of such shallow bound states, which is enforced by the term with ξ q in Eq. (3.28). Combined with the fact that the Coulomb kernel K is sharply peaked around q = k, this allows us to count all momenta from p0 and extend all integrations over the magnitude of such reduced momenta to infinite limits, as well as substitute 37 q−k q−k K(q, k) → K p0 + , p0 − ≡ K(q − k). 2 2 (3.30) This implies that Eq. (3.28) maps onto a 1D Schrödinger equation, in which 2ξ q = 2κ (q − p0 )2 + 2Λ → −2κ∂2x + 2Λ plays the role of an effective 1D kinetic energy (shifted by 2Λ, as we count the energies from the chemical potential); − gp0 δ( x ) represents the short-range attractive potential; K( x ) - the Fourier transform of K(q), is a long-range repulsive potential. As demonstrated above, for relatively small q p0 , which participate in the pairing problem at realistic values of parameters, one can approximate K(q) with (3.17). Here below we find an approximate corresponding 1D real space potential U ( x ) as the Fourier transform of the approximate expression of calK (q) given in (3.17). The Fourier transform is Z ∞ dq iqx e K( p0 + q/2, p0 − q/2). U (x) = −∞ 2π (3.31) By integrating by parts, this integral is reduced to U (x) = − 1 ix Z ∞ dq iqx d e K( p0 + q/2, p0 − q/2). −∞ 2π dq (3.32) At this point, one can substitute K( p0 + q/2, p0 − q/2) → K(q), which is now appropriate, since it yields a convergent result. This way, we obtain U (x) = Ec 2| x | exp − . 2π p0 | x | βd (3.33) This potential defines the behavior of the effective 1D quantum-mechanical problem down to distances | x | & 1/p0 . At smaller scales, the large-momentum behavior of K(q, p) becomes important, and Eq. (3.30) ceases to hold. From now and until the end of this section, we will measure all energies in units of ER , all momenta in units of p0 , all coordinates in units of 1/p0 , and g in units of κ. Then the effective Schrödinger equation in real space is written as −2α00 ( x ) − gδ( x )α( x ) + Ec − 2βd|x| e α( x ) = ( E − 2Λ)α( x ). 2π | x | (3.34) To qualitatively understand the solutions of Eq. (3.34), we start with the d → ∞ limit when e |x| − 2βd → 1. First, we focus on finding the critical value of g, gcb , at which the first bound 38 state emerges below the continuum – this signals the appearance of a two-particle bound state in the original problem. To find gcb , we set E = 2Λ, and retain only the solution that does not grow indefinitely for x → ∞. At distances | x | & 1, such solution is ! r q Ec | x | α 0 ( x ) = A | x | K1 , π (3.35) where A is an arbitrary constant, and K is the modified Bessel function of the second kind. Note that the asymptotic behavior of this function at large values of the argument is given by r K1 ( z ) ∼ π −z e . 2z (3.36) This implies that even though the state α0 has energy infinitesimally below the continuum, p it has a finite spatial extent α( x ) ∼ exp(− | x |/ζ 0 ), determined by ζ0 = π 1. Ec (3.37) (Note that in dimensionful units, ζ 0 = πER /( Ec p0 ).) The critical value of g is found from the condition that α0 ( x ) has a jump in derivative at the origin determined by g – the strength of the attractive δ-function – in the standard way. We note that for x → 0, the solution presented in Eq. (3.35) has a logarithmically divergent derivative. This divergence is unphysical, since the form of the Coulomb potential is not valid near the origin. Therefore, we cut-off the divergence at x ∼ 1, to obtain gcb ≈ 4|α00 ( x ∼ 1)| Ec 4π = ln 2 . α0 (0) π β Ec (3.38) It is clear that the critical gcb given by Eq. (3.38) also persists for finite d & ζ 0 . In turn, for 1 d . ζ 0 , the repulsive potential behaves qualitatively like a δ-functional one, simply renormalizing the strength of the attractive potential. That is, U (x) ≈ 2 Z d 1 2x 0 Ec dx exp − δ ( x ). 2πx 0 βd 0 (3.39) Performing the integral, and taking the d 1 asymptotics of the result, we observe that in this limit, the repulsive interaction renormalizes g by g → g− Ec d ln . π 2 (3.40) 39 The bound state appears when the renormalized strength of the δ-function potential is positive, hence in this regime gcb = Ec d ln . π 2 (3.41) Note that for d ∼ ζ 0 , Eqs. (3.38) and (3.41) are parametrically the same. Taken together, Eqs. (3.9), (3.38), and (3.41) determine the dependence of the critical attraction strength required to bind a two-particle bound state for all values of d. This dependence is schematically illustrated in Figure 3.4. The effective Schrödinger equation (3.34) can also help us determine the dependence of the bound state energy, Eb , on g > gcb . To this end, we write this equation for α0 , as well as for α g , the latter being assumed to solve the equation for g above the critical value, and take into account that 2Λ − Eb > 0 : − 2α000 ( x ) − gcb δ( x )α0 ( x ) + U ( x )α0 ( x ) = 0, − 2α00g ( x ) − gδ( x )α g ( x ) + U ( x )α g ( x ) = −(2Λ − Eb )α g ( x ). (3.42) We now multiply the first equation with α g , and the second one with α0 , subtract one from the other, and integrate over the entire axis. We immediately get 2Λ − Eb ( g) = R α0 (0) α g (0) ( g − gcb ). dxα0 ( x )α g ( x ) (3.43) The value of the prefactor in front of g − gcb depends on d and the value of g as compared to gcb . In two cases, either for d . ζ 0 and any g, or for g gcb , the size of the bound state p in the effective δ-potential is determined by its energy, ζ g ∝ 1/ ε cb ( g). Thus, we have q α (0) α g (0) R 0 ∝ 2Λ − Eb ( g), (3.44) dxα0 ( x )α g ( x ) and hence, 2Λ − Eb ∝ ( g − gcb )2 . (3.45) . In the limit of large d & ζ 0 , and for g − gcb gcb , we can estimate R α0 (0) α g (0) 1 ∝ , dxα0 ( x )α g ( x ) ζ 0 (3.46) and conclude that 2Λ − Eb ∝ ( g − gcb ). (3.47) 40 Figure 3.4. Critical gcb for the two-particle bound state formation in various regimes of screening. In principle, the boundary of the insulator-superconductor transition can be obtained by solving Eq. (3.43), and setting Eb ( g) = 0. We will not pursue the task of its determination, deferring it until Section 4.2, where many-body aspects of the condensation problem are discussed. The intuition used in that section, however, heavily rests on the results of this one. In the next chapter using a more sophisticated treatment that allows investigating behavior of the order parameter at zero temperature, we study the nature of the superconducting transition in more details. There, we extend the problem to the case of negative Λ < 0, when in the absence of attraction, the normal state is a shallow Fermi moat of electrons. We will show how in that case, a crossover between BEC-BCS regime can occur by changing parameters of the system and based on that will construct a phase diagram of the system in the absence and the presence of the Coulomb repulsion. CHAPTER 4 VARIATIONAL BCS WAVEFUNCTION In this chapter, we improve our treatment by using a BCS-type variational wavefunction. This allows us to investigate the situation with finite electron density in the moat band. It should be clarified here that by considering a BCS variational wavefunction, we are assuming that the normal state of the model at Λ < 0, i.e., when the chemical potential is inside the moat band, is a Fermi liquid one. This assumption is based on previous studies [19, 150] of the phase diagram of the repulsively-interacting electrons which found that other, more exotic, electron phases in this interesting regime – such as Wigner crystal, nematic, and ferromagnetic phases – require significant Coulomb repulsion (see, for example, Fig. 1 of Ref. [150]). Such a strong repulsion is not consistent with the superconducting state we are considering here. As shown below, the BCS wavefunction allows us to study the behavior of the superconducting order parameter ∆ across the crossover region between the BEC and BCS limits, in the absence and presence of Coulomb interaction. The variational wavefunction that we consider is |Ψ BCS i = ∏ (u p + eiθ p x >0 p v p c†p c†− p )|0i, (4.1) where u p and v p are the variational functions; v p is multiplied by the phase factor, which is chosen to compensate for the phase factors in the attractive channel, Eq. (2.8). Similar to Chapter 3, we first study the model without repulsion and then will investigate the effects of Coulomb repulsion. 4.1 Without Coulomb repulsion We start with summarizing the phase diagram in ( g, Λ) coordinates for the case of neutral fermions with attraction. It is depicted in Figure 4.1. Qualitatively, for g > 0, the phase diagram contains two phases: an insulator, and a 42 Figure 4.1. Phase diagram in the absence of Coulomb repulsion. gc line on the Λ positive side is drawn based on Eq. (3.5). superconductor. The superconducting phase is stable for all values of Λ ≤ 0 (chemical potential is inside, or touches the majority band), and for g > gc (Λ) for positive Λ (when the chemical potential is below the majority band); see Figure 4.1. For Λ > 0, the transition from the insulator into a superconductor upon increasing g occurs into a BEC-type superconducting phase, which is represented by a dilute gas of bound pairs of electrons. For Λ ≤ 0, the superconducting phase has BCS character, in which it is no longer permissible to disregard the fermionic nature of the electrons comprising the pairs in the condensate and think of it as a dilute Bose gas. There are two notable lines in the phase diagram of Figure 4.1: the line of superconductorinsulator transition, and the line of BEC-BCS crossover. The superconductor-insulator transition line is determined from the condition that the bound state energy for electronic pairs lies at the chemical potential for pairs (chosen to be zero in our analysis). In other 43 words, counted from the bottom of the two-electron continuum, the binding energy of the pair must be equal to 2Λ. The corresponding critical value of g was calculated in Section 3.1, and is given by Eq. (3.5). The BEC-BCS crossover line has the inherent vagueness in its definition, which depends on the criterion for the crossover. Here we adopt the following logic for drawing such lines: the BEC-BCS crossover is a crossover between regimes in which the Fermi statistics of pair constituents either does not or does matter. The importance of carrier statistics is dictated by the occupations of states in momentum space, so we define the crossover as the situation in which the maximum occupation of states in the momentum space is 1/2. It will become apparent below that for the case of neutral fermions, and within the model adopted in the present paper, the BEC-BCS line is then the vertical Λ = 0 one – the g-axis. To get further quantitative insight into the behavior of the superconducting order parameter, we study its variation along a horizontal cut of the phase diagram in Figure 4.1, that is, we vary Λ at a given value of g. We will show below that the superconducting gap, ∆(Λ), is represented by a universal curve, which we determine below. Minimization of (2.1) and (2.8) over the BCS wavefunction (4.1) leads to the standard self-consistent equation [89] for superconducting order parameter ∆ 1= g 2V ∑q p 1 ξ 2p + ∆2 , (4.2) where ξ p is the single electron dispersion (2.4). Integrating over the angle and noting that the most important part of the radial integral comes from the vicinity of p0 , we approximate pdp ∼ p0 dp and extend the p-integral to ±∞. This gives 1 1= π Z ∞ −∞ dx q ( x2 + Λ̃)2 + ∆˜ 2 , (4.3) ˜ = ∆/Λc0 are made dimensionless with the help of Λc0 = where Λ̃ = Λ/Λc0 and ∆ g2 p40 /16ER . This integral equation holds for both positive (chemical potential below the band) and negative (chemical potential inside the moat band) Λ̃. Note that the characteristic energy Λc0 scales up like g2 . It is clear that for reasonably small g, Λc0 stays much smaller than the Rashba energy ER . Therefore, Λ̃ can be large even in the ‘deep moat’ limit | Λ | ER . 44 ˜ (Λ̃) of Eq. (4.3). Finite ∆ ˜ for positive Λ̃ describes Figure 4.2 shows numerical solution ∆ BEC regime - condensed state of two-electron pairs. The order parameter vanishes at Λ = ˜ Λc0 with a discontinuity in its derivative. It can be shown that asymptotic behavior of ∆ q ˜ ∼ 8 (1 − Λ̃). near the transition point is ∆ 3 ˜ (Λ̃) (Figure 4.2) is that it has a maximum on the negative The other feature of function ∆ side, when the chemical potential is inside the moat band. That implies that the highest critical temperature happens when the chemical potential is slightly above the band edge. Moreover, on the BCS side, when electrons fill up a shallow ring-shaped Fermi sea, the ˜ has two distinct behaviors: when |Λ| is of the order of Λc0 , the order order parameter, ∆, parameter is of the same order and is proportional to g2 . For larger |Λ|, when |Λ̃| 1, the p order parameter decays exponentially as ∆ ∝ |Λ| exp − π2 |Λ|/Λc0 . To see this, expand 2.00 1.75 B 1.50 A ˜ ∆ 1.25 1.00 0.75 0.50 0.25 0.00 -15 -10 -5 Λ̃m 0 1 Λ̃ ˜ (Λ̃). Two important points of the function are shown Figure 4.2. The universal function ∆ in the figure: A is the point where the chemical potential touches the band edge, which is ˜ 0 ] ≈ [0, 1.393]; B is the point the point of the BEC-BCS crossover, with coordinates [Λ̃0 , ∆ ˜ is located, with coordinates [Λ̃m , ∆ ˜ m ] ≈ [−1.424, 1.655]. where the maximum of ∆ 45 ( x2 − |Λ|)2 ≈ 4|Λ|( x − |Λ|1/2 )2 and approximate the integral in (4.3) by the region around x ≈ |Λ|1/2 . The large-|Λ| has a familiar form exp(−const/g) for the order parameter in a standard BCS theory, suggesting a superconducting state with strongly overlapping electron pairs at large enough |Λ|. Finally, the BEC-BCS crossover line can be deduced from the usual BCS coherence factor, v p , whose square defines the occupation of states in the momentum space:  v2p =  ξp 1 . 1− q 2 2 2 ξp + ∆ (4.4) It is clear that for Λ > 0, and hence ξ p > 0, all occupation numbers are smaller than 1/2. In turn, for Λ < 0, when ξ p changes sign at the Fermi momenta, there is a region in momentum space – where there is “water” (i.e., electron liquid) in the moat – where the occupation numbers exceed 1/2. For Λ = 0, regardless of g, the maximum occupation is exactly 1/2, and is reached at p = p0 , at the bottom of the moat. Hence, the Λ = 0 is the BEC-BCS crossover line according to the criterion we chose, as shown in Figure 4.1. 4.2 Effects of Coulomb repulsion The inclusion of Coulomb repulsion between electrons changes the phase diagram, which is now shown in Figure 4.3. It is convenient to describe phases appearing in this diagram starting with the behavior of the system along Λ = 0 line. As we know from Section 3.2, in the presence of the Coulomb repulsion, there is a finite value of g, which we denoted with gcb , needed to achieve binding of pairs even at Λ = 0. Therefore, for 0 ≤ g < gcb , the Λ = 0 line is the boundary between the insulating phase on the right, and the normal metal phase on the left. In other words, the system behaves as a semi-metal with a diverging one-dimensional density of states for g < gcb , and Λ = 0. To the left of this semimetallic phase, there lies a normal metal phase. The doping level of this metal is lower than the nominal one prescribed by |Λ|, due to electrostatic effects. Superconducting phases can be reached from both metallic and insulating phases. As in the case of neutral fermions, one reaches BEC- or BCS-like states increasing g on the Λ ≷ 0, respectively. Exactly at Λ = 0, one goes into a BEC superconductor, since the BEC-BCS line lies on the left of the g-axis in this case; see below. 46 Figure 4.3. Phase diagram in the presence of Coulomb repulsion. The boundaries were found numerically for a weakly screened Coulomb case with parameters: dp0 = 1000 and Ec = ER /10. Λeff on the x-axis corresponds to the value of Λ + Φ in the normal state, which is the same as bare Λ on the insulator side and is renormalized strongly on the metallic side according to Eq. (4.21). Having described the qualitative features of the phase diagram for charged fermions, we elaborate now on its quantitative aspects. The Coulomb repulsion brings two new terms to the total energy, one through the Cooper channel (third term in the equation below) and the Hartree-Fock term (forth term below), to which the attractive interaction gives no contribution, E = EK + E g + Ec,∆ + EH-F = ∑ v2k ξ k − k + 1 2V 1 ∑0 uk vk uk0 vk0 Ũk,k0 + 2V ∑0 v2k v2k0 kk kk g 2V ∑ u k v k u k0 v k0 kk0 Ũ0 − Ũk,k0 , (4.5) where Ũk,k0 = Uk−k0 (1 + cos(θk − θk0 ))/2, as before. Minimizing the energy over vk = sin αk (with uk = cos αk ) and introducing fields 47 ∆k = Φk = 1 V ∑0 uk0 vk0 1 V ∑0 v2k0 (Ũ0 − Ũk,k0 ), k g − Ũk,k0 ), (4.6) k we obtain desired coupled self-consistent equations for the superconducting and HartreeFock (HF) fields 1 ∆p = 2V ∑0 q p ∆ p0 g − Ũ p,p0 (4.7) (ξ p0 + Φ p0 )2 + ∆2p0  1 Φp = 2V ∑0 1 − q p ξ p0 + Φ p0 (ξ p0 + Φ p0 )2 + ∆2p0   Ũ0 − Ũ p,p0 . It is clear from Eqs. (4.7) that the Coulomb repulsion has a two-fold effect on the pairing problem. First, it changes the coupling constant in the equation for the order parameter ∆ p , g → g − Ũ p,p0 , making it a coupling matrix. Physically, this describes the interaction of electrons within a pair, and the effects of this interaction have been described in Section 3.2. Second, the Coulomb repulsion leads to the appearance of the Hartree-Fock renormalization of the single-particle spectrum, described by Φ p . The first bracket in the equation for Φ p is just 2v2p0 , twice the particle number at the momentum p0 . Moreover, the second bracket in the same equation is always positive because Coulomb interaction (2.12) is maximum at p = 0, Ũ0 = e2 d/2e. Therefore, the new collective field Φk is a kind of k-dependent chemical potential, which provides a positive shift of the single particle spectrum ξ k . This is the same as downward renormalization of the chemical potential and has the effect of ‘pushing out’ electrons from the system. The renormalization is proportional to the Coulomb repulsion experienced by a particle at momentum k due to all other particles (at momenta k0 ). In a usual good conductor, such a renormalization is neglected due to the strong screening of Coulomb interaction. This point is made explicit in classic papers on strongly coupled superconductivity; see for example [156, 159] (note that Φ is denoted as χ there). The reason is that renormalization of the spectrum, which represents a “simple scale change” (Ref. [156]) of electron dispersion, is a small effect on the scale of Fermi energy EF , which is present in both normal and superconducting states. In our case, the normal state is either a band insulator, or a metal with a very small carrier density, hence the scale change due to 48 Φ is crucial in determining whether or not electrons can populate the bottom of the moat band, either in the form of interacting electron gas or bound two-electron molecules. Therefore, the combination of low-density and weak screening (which is determined by the distance due to the external gate) results in the HF field Φ p playing an unusually important role. The basic physics is that of electrostatic repulsion between two electrons attempting to form a bound state. In the absence of screening and retardation (our attractive potential g is frequency independent constant), the electrostatics of the pair becomes crucial. As the discussion in Section 4.2.2 below shows, weakly screened Coulomb interaction strongly renormalizes bare chemical potential (see Eq. (4.21) below) and squeezes electrons out, providing a kind of ‘Coulomb blockade’ phenomenon in a macroscopic setting. This crucial physics is missing in Ref. [76]. It is clear from Eq. (4.7) that the presence of the Coulomb interaction leads to p-dependence of both ∆ p and Φ p . However, it is not so clear if the possible solutions are radially symmetric or not. We have carefully investigated this question numerically and concluded that the lowest energy solution of the coupled equations (4.7) is s-wave symmetric. Therefore, below we investigate solution with s-wave symmetry and perform angular integration to obtain one-dimensional integral equations as follows: ∆p = Φp = Z p0 max p 0 ∆ p0 Ec 0 g − K( p, p ) , p0 0 (ξ p0 + Φ p0 )2 + ∆2p0   Z p0 max ξ p0 + Φ p0 Ec  d − K( p, p0 ) , dp0 p0 1 − q 4π p0 0 (ξ p0 + Φ p0 )2 + ∆2p0 1 4π 0 dp q (4.8) where the Coulomb kernel K( p, p0 ) is given by Eq. (3.14). The upper momentum cut-off, pmax , is taken to be 2p0 in numerical solutions of the above equations. Armed with Eqs. (4.8), we are going to consider the limits of strong (dp0 1) and weak-screening (dp0 1), similar to Chapter 3. 4.2.1 Strong screening We start with the limit of strongly screened Coulomb interaction. In this limit, K( p, p0 ) ≈ Ec d . 2p0 (4.9) 49 With this approximation, the p-dependence of ∆ and Φ disappears, and the effect of Coulomb interaction is to reduce g by Ec d/2p0 (compare with (3.10)), resulting in an effective attraction geff = g − Ec d . 2p0 (4.10) 2 p4 /16E , introduce Following the notation used in Section 4.1, we define Λc,eff = geff R 0 dimensionless combinations Λ̃ = Λ/Λc,eff and Φ̃ = Φ/Λc,eff and extend integration limits due to the fast convergence of the integral. The first equation in (4.8) becomes 1= 1 π Z ∞ −∞ dx q ( x2 + Λ̃ + Φ̃)2 + ∆˜ 2 , (4.11) which is just the same as Eq. (4.3) upon switching Λ̃ + Φ̃ to Λ̃. The second equation in Eq. (4.8) can be written as Φ̃ = g − geff πgeff Z ∞ −∞  1 − q x2 ( x2  + Λ̃ + Φ̃ + Λ̃ + Φ̃)2 + ∆˜ 2  dx. (4.12) ˜ in terms of Λ̃ + Φ̃ is given by the same universal function that Eq. (4.11) shows that ∆ is plotted in Figure 4.2. Therefore, with appropriate rescaling Λ̃ → Λ̃ + Φ̃, one can find dependence of ∆ on the bare Λ. For instance, from Figure 4.2, we know that the order ˜ turns to zero at the critical point Λ̃ + Φ̃ = 1. Eq. (4.12) shows that at this parameter ∆ point, Φ̃ is also zero, and therefore we conclude that Λ̃ = 1, that is Λ = Λc,eff , is the bare critical Λ in this case too. The same logic shows that the maximum value of ∆ does not depend on the renormalization of Λ and is given by ∆m ≈ 1.65Λc,eff . Finally, similarly to Section 4.1, there too is a region where |Λ̃ + Φ̃| Λc,eff and the order parameter is exponentially small. If g/geff − 1 is very small, the asymptote in terms of bare Λ has the same form as in the case with no Coulomb interaction, because then Λ̃ + Φ̃ ∼ Λ̃. In the other limit, when geff g, although we get a different asymptote as far as Λ-dependence is concerned, we still get a similar g-dependence in ∆ that behaves as ∆ ∼ exp(−const/geff ). To summarize, the case of strongly screened Coulomb interaction is very similar to that with no Coulomb interaction. Λ + Φ plays the role of effective Λ where, thanks to the strong screening, Φ slightly modifies the bare Λ. 50 4.2.2 Weak screening Now we consider the weak screening limit where the distance to the screening gate d is large in comparison with 1/p0 , Coulomb interaction is only weakly screened, and therefore repulsion between electrons is strong. From the results of Section (3.2), it is apparent that even in the dp0 1 case, one should distinguish two regimes: d . ζ 0 ≡ πER /Ec p0 , and d & ζ 0 . As we saw in Section (3.2), ζ 0 plays the role of the size of the two-particle bound state on its appearance for d → ∞. In the d . ζ 0 regime, which can be called the regime of intermediate screening, one can still introduce an effective local coupling constant; see Eq. (3.40). While the electrostatic effects, described by Φ p , are strong in this case, the basic physics is very similar to the strong screening case. Therefore, in what follows, we concentrate on the d & ζ 0 limit of really weak screening. We start the analysis of Eqs (4.8) with noting that the p-dependence of the Φ field can be neglected. To see that, note that for large dp0 , the maximum of function K( p, p0 ) goes like ln(d); see Eq. (3.15) of Chapter 3. Therefore, the K-term in the second bracket in the equation for Φ (4.8) can be dropped in comparison with the first, d term, and the Φequation simplifies to Φ= Ec d n(∆, Φ, Λ), p0 (4.13) where the particle density n is given by n= Z p0 ∞ 4π −∞  1 − q  ξp + Φ (ξ p + Φ )2 + ∆2p  dp. (4.14) Next, let us assume that ∆ p ξ p + Φ. We are going to show that this assumption holds as long as the pair condensate is dilute. Under this assumption, the first of Eqs. (4.7) can be linearized as 1 ∆p = 2V ∑0 p ∆ p0 g − Ũ p,p0 . ξ p0 + Φ (4.15) We then notice that the above equation is equivalent to the single pair Schrödinger equation (3.7) upon substituting the following ansatz for ∆ p : ∆ p = δ(ξ p + Φ)α p , (4.16) 51 where δ is a normalization constant to be found from Eq. (4.13). The proposed above ansatz works if one identifies Φ=− Eb . 2 (4.17) The condition (4.17) requires Eb < 0, since Φ always has a positive value proportional to the density (see Eq. (4.13)). In other words, we can say that if the single pair Scrödinger equation (3.7) has a solution with E < 0, then through (4.16) and (4.17), one can construct a solution for the linearized BCS equation (4.15) out of it. Note that for the case of Λ > 0, such a solution for (3.7) is automatically a bound state. That is because for Λ > 0, the bottom of the continuum is always above the chemical potential. We therefore conclude that Φ = − Eb /2, where, as before, Eb is the two-particle bound state energy counted from the chemical potential for pairs. Since on the Λ > 0 side Φ 6= 0 corresponds to the presence of a superconducting condensate, we conclude that Eb ( g, Λ) = 0 (4.18) is the equation for the superconductor-insulator transition line, gc (Λ). Therefore, it can be obtained from the solution of the two-particle problem, described in Section 3.2. The result of numerical solution of this problem is shown in Figure 4.3, on the Λeff > 0. Now we show that our approximation, ∆ p ξ p + Φ, equivalent to δα p 1, is consistent as long as the condensate is dilute. To see that, consider the ansatz (4.16) and use the approximation, ∆ p ξ p + Φ, to write the density (4.14) as n≈ p0 2 δ 8π Z α2p dp, (4.19) where δ 1, as appropriate for the onset of superconductivity, and α p is a normalizable bound state wavefunction of the single pair Schrödinger equation (3.7). This implies that α p is peaked around p0 , and has a finite width proportional to the inverse of the pair-size, R ζ. Hence, one can estimate α2p dp ∼ α2p0 /ζ. Applying this to Eq. (4.19), we conclude that in order for the ansatz Eq. (4.16) to be valid, we must have δ2 α2p0 ∼ nζ 1. p0 (4.20) At the point of transition, g = gc , the bound state energy of a single pair, as well as density-dependent quantities Φ and n, are zero, while the pair has a finite size for Λ > 0, 52 which means (4.20) is valid at the transition point. As g increases above gc , the HF field Φ and the density n both increase with g − gc , while the pair-size slowly shrinks. Therefore, we conclude that in the case of Λ > 0, one can think of the superconducting state as a dilute condensate of well-defined pairs as long as the system is in the vicinity of the transition point. Figure 4.4 shows that the solution constructed by the ansatz (4.16) matches the numerical solution of Eq. (4.8) very well. Some insight into the shape of the gc (Λ) line for the superconductor-insulator transition on the Λ > 0 side of the phase diagram can be deduced from Eqs. (3.45) and (3.47). Setting Eb = 0 in those equations, we see that in the vicinity of Λ = 0, the transition line √ has a finite slope, gc − gcb ∝ Λ, while for gc − gcb & gcb , we have gc − gcb ∝ Λ. Both of these observations are confirmed numerically in Figure 4.3. Next we consider the case Λ < 0, when the bare chemical potential is inside the band. This situation is illustrated in the insert of Figure 2.2. For g = 0, one has a gas of interacting electrons, with density strongly reduced from the nominal value for the noninteracting case due to the strong electrostatic effects. We can determine the normal-state density by noting that since ξ p + Φ = κ ( p − p0 )2 − |Λ + Φ| in Eq. (4.14) changes sign as a function of p, there is a finite electron density even in the absence of superconductivity, ∆ p = 0. For d ζ 0 , solving Eqs. (4.13) and (4.14) together leads to |Λ + Φ| ≈ ζ 02 Λ2 | Λ |. d2 ER (4.21) This shows that negative bare Λ causes positive Φ, the magnitude of which is comparable to |Λ|, so that the renormalized Fermi energy of the electrons counted from the bottom of the band, |Λ + Φ|, is much smaller than the bare one, |Λ|. The Coulomb repulsion pulls the effective chemical potential down (decreases |Λ + Φ|) so as to reduce the density of electrons. We can define a normal-state Fermi momentum, counted from p0 , associated with the normal state Fermi energy, |Λ + Φ|: s |Λ + Φ| p0 ζ 0 | Λ | p F = p0 ≈ . ER d ER (4.22) Qualitative features of the superconducting transition on the Λ < 0 side depend on the relation between p F and 1/ζ 0 , since 1/ζ 0 determines the spread of momenta around p0 that participate in the formation of the two-particle bound state. For p F . 1/ζ 0 , the 53 ∆p /ER 6 ×10−6 4 2 iteration ansatz 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 1.00 1.25 1.50 1.75 2.00 ×10−4 ∆p /ER 1.9 1.8 1.7 iteration ansatz 1.6 0.00 0.25 0.50 0.75 p/p0 Figure 4.4. Comparing numerical solution of Eq. (4.8) (labeled as iteration) and the ansatz (4.16) (labeled as ansatz) for Λ = 0 (top panel) and Λ = ER /100 (bottom panel). g is chosen to be slightly above gc , such that the diluteness condition (4.20) holds. For the top panel, g/κ = 0.175, and for the bottom panel, g/κ = 0.52. In both cases, screening is very weak: dp0 = 1000 and Ec = ER /10. In order to make the ansatz solution, the Schrödinger equation (3.7) has been solved numerically to find α p and then using Eq. (4.17) and (4.13) the normalization constant δ was found. transition is strongly affected by the formation of the bound state on the BEC side. In that case, one expects ∆ p with a strong momentum dependence, whose form is largely similar to the one obtained for Λ = 0 at large momenta, | p − p0 | & p F (see Figure 4.5), with some modifications around the normal-state Fermi surface. In this regime, we expect a value slightly less than gcb for gc , the reduction being due to the increased right-hand side of the self-consistency equation – the first of Eqs. (4.8) – in the presence of zeros of ξ p . For p0 54 p F & 1/ζ 0 , the states that participated in the formation of the bound state are completely Pauli-blocked, and BEC physics is irrelevant. In this limit, one expects the pairing physics to be dominated by the Fermi surface, with only the value of ∆ p at the Fermi momentum being important. Since we are interested in the BEC-BCS crossover physics, we do not pursue the problem of determining the normal-superconducting phase boundary in full rigor and present only numerical results here; see Figure 4.3. Of importance to us is the undoubtful fact that the BCS phase does exist on the Λ < 0 side and the critical gc (Λ) separating the normal and superconducting phases lies below the BEC-BCS crossover line. This latter line will be described below and is precisely determined numerically. To see how the ∆ p solutions of Eq. (4.8) behave on the Λ < 0 side, we present a typical numerical solution for Λ = − ER in Figure 4.5. The lowest panel corresponds to the smallest g where Λ + Φ < 0 and we are in the BCS region; the middle panel belongs to the boundary case where Λ + Φ = 0, which corresponds to the BEC-BCS crossover, see below; and finally, for an even larger g in the top panel, Λ + Φ becomes positive and we are in the BEC region. Notice how in the BCS region, ∆ p has two very sharp dips on the Fermi momenta and as we go into the BEC region, it becomes more like the solutions on the positive side of Λ presented in Figure 4.4. It is also noteworthy that we are still within the validity of the moat-band model even for Λ = − ER , since the normal-state doping level is much smaller than |Λ|; see Eq. (4.21). This tendency is seen in Figure 4.5: while the bare value of Λ = − ER in all three cases shown there, the renormalized values of |Λ + Φ| are indeed much smaller than |Λ|, and are in agreement with Eq. (4.21). In Figure 4.6, the numerical solutions of Φ p are also plotted for the same set of parameters. It can be seen how its p-dependence is more pronounced in the case with the smallest g. Nonetheless, by looking at the y-axis scale, it’s clear that the p-dependence is negligible in all cases. The BEC-BCS crossover line can be deduced from the same considerations as in the case of neutral fermions, except the condition Λ = 0 should be replaced with Λ + Φ( g, Λ) = 0, which implicitly defines a line in the ( g, Λ) plane. It is clear that this line exists only on the Λ ≤ 0 side of the phase diagram, and starts from point ( gcb , 0). The rest of the line can be obtained numerically in the following way: first, one sets Λ + Φ = 0 in the equation for ∆ p , and thus obtains a solution for every g > gcb . Having obtained a solution for ∆ p 55 1.3 ×10−2 1.2 Λ + Φ = 3.43 × 10−3 ER g/κ = 0.5 1.1 1.0 ∆p /ER 7 ×10−3 6 Λ+Φ=0 g/κ = 0.358 1.5 ×10−4 1.0 0.5 0.0 0.00 Λ + Φ = −9.66 × 10−4 ER g/κ = 0.165 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p/p0 Figure 4.5. Typical ∆ p numerical solution of Eq. (4.8) for Λ = − ER and very weakly screened Coulomb: dp0 = 1000 and Ec = ER /10. The lowest panel is a typical solution of the BCS kind while the top one was found in the BEC region. The middle panel represents the boundary case where Λ + Φ = 0. For each panel, numerical values of g and Λ + Φ are written on the plot. for a particular g, one then substitutes this solution into the equation for Φ, Eq. (4.13), in which Φ itself in the left-hand side is replaced with |Λ|. As a result, one obtains |Λ( g)| that defines the BEC-BCS crossover line. The result of implementing this program is shown in Figure 4.3 as a dashed line on the Λeff < 0 side. This concludes the construction of the phase diagram for the case of charged fermions. This completes our analysis in this chapter. In the next chapter, we are going to employ a path integral approach that will let us study the system at finite temperature. The action method also provides a framework to study fluctuations of the system. 56 1.004 1.003 Φp /ER 1.002 1.001 g/κ = 0.165 g/κ = 0.358 g/κ = 0.5 1.000 0.999 0.998 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p/p0 Figure 4.6. Φ p numerical solution of Eq. (4.8) for Λ = − ER and very weakly screened Coulomb: dp0 = 1000 and Ec = ER /10. Each curve is labeled by its corresponding g/κ value, while the same set of g is used as in Figure 4.5 CHAPTER 5 ACTION TREATMENT In this chapter, we treat the problem using the action method. Based on the Hamiltonian made in Chapter 2, we are going to construct its corresponding action to be used in the path integral approach that we take here. We show how some of the results achieved by the two other approaches in Chapter 3 and Chapter 4 can be reproduced by the action method. As we briefly outline below, using imaginary time, one can express the partition function of a system in terms of a path integral. Partition function makes it possible to study the statistical properties of the system. It enables us to carry on some finite temperature analysis that is not accessible through wavefunctions. As we see below, the superconducting order parameter will show up as an imaginary auxiliary bosonic field. The other advantage of the approach that we use here is that it also provides us with a framework to study the effects of fluctuations on that superconducting field. 5.1 Building the action We are used to this idea that in Quantum Mechanics, observables are represented by operators. Operators could be position or momentum of a particle or, as it’s the case in the context of many body systems, field operators. Path integral approach leaves this picture and uses numbers instead of operators. In this approach, we make an action out of appropriate numerical values and then consider all possible paths or configurations these values can take. There is a standard way to go from a many-body Hamiltonian made of field operators to an equivalent action that is made of numerical values. The procedure and derivation of it can be found in many textbooks on the topic of Quantum Field Theory such as Ref. [120] by Nagaosa – a concise well-written book that we follow here. 58 5.1.1 General concepts In order to outline the idea of path integral approach and its connection to partition function, consider a Hamiltonian H and its complete set of states labeled as | x i with appropriate numbers. Then it can be shown that for a propagator that connects two different states | x i and | x 00 i at time 0 and t, we have h x 00 |e−iHt | x i = Z x,x 00 D x 0 (t0 ) exp −S{ x 0 (t0 )} , where S is the corresponding action to H and the symbol R x,x 00 (5.1) D x 0 (t0 ) represents integrat- ing over all possible paths that connect the states | x i at time zero to the state | x 00 i at time t. On the other hand, we know from Statistical Mechanics that for a quantum mechanical system, canonical partition function is given by (see any Statistical Mechanics text book such as Ref. [131]) h i Z Z = Tr e− βH = dx h x |e− βH | x i, (5.2) where β = 1/k B T is inverse temperature. Comparing (5.2) and (5.1), one should realize that if we introduce imaginary time as t0 = −iτ, set the final time t = −iβ, choose the starting and the ending states to be the same, and then integrate over all possible starting sates, partition function can be written as the following path integral Z= Z h i D x e−S{ x } . (5.3) Therefore, to express the partition function of a system as a path integral, all we need is to know how to make the system’s action based on its Hamiltonian. It can be shown that for both bosonic and fermionic fields, such correspondence is given by the following recipe (see Ref. [120]) S= ∑ p Z β 0 dτ ψ̄ p (τ )∂τ ψ p (τ ) + H (ψ̄ p (τ ), ψ p (τ )) , (5.4) where p goes over all states of the Hamiltonian H. If c†p is the creation operator at state p and c p is the annihilation operator at the same state, ψ̄ p and ψ p are the corresponding numbers that replace them. Following the idea explained above, the imaginary time τ spans between the starting state and the ending state in the underlying propagator. But in the partition function relation (5.2), the starting sate and the ending state are the same. 59 This sets a boundary condition on ψ̄ p (τ ) and ψ p (τ ). The boundary condition depends on whether the field is bosonic or fermionic. If it’s a bosonic field, the boundary condition is ψ p ( β ) = ψ p (0), ψ̄ p ( β) = ψ̄ p (0). (5.5) This periodicity in τ-space means that the bosonic ψ and ψ̄ can be expressed by Fourier series with the following discrete set of frequencies ωn = 2nπ = 2nπk B T. β (5.6) For the fermionic fields, the boundary condition is anti-periodic ψ p ( β ) = − ψ p (0), ψ̄ p ( β) = −ψ̄ p (0). (5.7) Which corresponds to the following set of discrete frequencies in their Fourier series ωn = (2n + 1)π = (2n + 1)πk B T. β (5.8) ωn in (5.6) and (5.8) are known as Matsubara frequencies for bosonic and fermionic systems, respectively. The crucial step in the recipe (5.4) is replacing the field operators with numbers. In the case of fermionic fields, such as electrons, this gets a little tricky. In fact, in this case, the replacement cannot be done using real or complex numbers. Luckily, in the world of mathematics, a suitable set of numbers to represent fermionic fields had been already developed, before physicists needed them. They are called Grassmann numbers, and they have properties similar to fermionic operators. For instance, they anti-commute, i.e., for two different Grassmann numbers, we have gi g j + g j gi = 0, (5.9) which immediately yields that gi2 or any higher integer power of a Grassmann number is zero. This means that any analytic function of a set of Grassmann numbers can be written as a simple series with no higher powers. For instance, if g1 and g2 are Grassmann numbers, we have exp( g1 ) exp( g2 ) = 1 + g1 + g2 + g1 g2 . The other essential element in Grassmann calculus is the special notion of Grassmann integral, which is defined as Z gi1 · · · gin dg jn · · · dg j1 = e, (5.10) where e is either 0 or ±1. If the set of {i1 · · · in } is not identical to { j1 · · · jn }, it is 0. If the two sets are identical and also have the same order or can be made in the same order by 60 even number of permutations, then it is +1; if the required number of permutations is odd, it is −1. Using these rules and notions, it can be shown that an equivalent formula for the general form of Gaussian integrals exists for a set of Grassmann numbers like ψ̄i and ψj as follows Z " exp − ∑ ψ̄i Aij ψj # i,j n ∏ dψ̄i dψi = det A, (5.11) i =1 where A is a square matrix. The famous counterpart of (5.11) for complex numbers is given by Z " exp − ∑ zi∗ Aij z j # i,j n ∏ dzi∗ dzi = i =1 πn . det A (5.12) Note that each complex number z has two dimensions, which means the integral in (5.12) is a 2n-dimensional integral. The Gaussian integral formulas (5.11) and (5.12) are the most important tools in dealing with path integral calculations. A more extensive and detailed material on Gaussian integrals that appear in path integral calculations can be found in the work of Hagen Kleinert in Ref. [95]. And the standard reference book that covers the topic of Grassmann numbers and their analysis is Ref. [47] written by Bryce DeWitt. 5.1.2 Making the effective action Now following the recipe (5.4) and using the projected Hamiltonian built in Chapter 2, we proceed to make the proper action of our model. The noninteracting part of the Hamiltonian, the moat band of H0 in (2.1), gives us the noninteracting part of the action as S0 = ∑ p Z β 0 dτ ψ̄p (τ )∂τ ψp (τ ) + ξ p ψ̄p (τ )ψp (τ ) , (5.13) where ψp and ψ̄p are Grassmann numbers since our system is of course an electronic system. Next, we go from the imaginary time, τ, space to Matsubara frequency space. To do that, we introduce the following Fourier transform 1 ψp (τ ) = p e−iω τ ψ(ω ,p) , ∑ β n n ωn 1 ψ̄p (τ ) = p eiω τ ψ̄(ω ,p) , ∑ β n n (5.14) ωn where ωn are fermionic Matsubara frequencies defined in (5.8). It is convenient to introduce a four-vector as l = (ωn , p). Then using the four-vector index, l, and going to Fourier space, S0 becomes 61 S0 = ∑(−iωn + ξ p )ψ̄l ψl , (5.15) l which is now simpler than (5.13) and has a clear quadratic form. A familiar reader with Green’s functions in the context of propagators can recognize in (5.15) the presence of a Matsubara Green’s function of free electrons in the band ξ p . The free Matsubara Green’s function is defined as Gl = 1 , iωn − ξ p (5.16) and is going to be the building block of all the forthcoming calculations. Next, we consider the contribution of the Coulomb repulsion in the action. Starting from Hc given in (2.9) and performing Fourier transform to go from τ-space to frequency space, we obtain Sc = 1 Ueff (l, l 0 , m)ψ̄l +m ψ̄l 0 −m ψl 0 ψl , 2Vβ ll∑ 0m (5.17) where we used the four-vector indices and Ueff is defined in (2.10) with the same gated Um that is given in (2.12). For the other interaction term in our model, i.e., for the Cooper channel contact attraction H p given in (2.8), we introduce a small modification; we add a new degree of freedom. H p in (2.8) assumes that pairs are sitting exactly at the opposite momenta. Here we are going to let them wiggle a little bit around the Cooper channel; that is instead of putting them at p and −p, now we put them at p + Q/2 and −p + Q/2 so that the total momentum of a pair is Q instead of zero. Q which is assumed to be a small deviation from zero is the new degree of freedom. It essentially allows for long wavelength fluctuations on top of the emerging superconducting field. One can see that with this modification, the attractive part of our action after Fourier transform and using four-vector indices would be written as Sp = − g (e−iθl+n/2 +iθl0 +n/2 + e−iθl−n/2 +iθl0 −n/2 )ψ̄l 0 +n/2 ψ̄−l 0 +n/2 ψ−l +n/2 ψl +n/2 . 4Vβ ll∑ 0n (5.18) Now that we prepared all three parts of the full action S = S0 + Sc + S p , given by (5.15), (5.17), and (5.18), respectively, following (5.3), the partition function shall be given by the following path integral Z= Z h i D ψ̄D ψ e−(S0 +S p +Sc ) . (5.19) If the action only had the noninteracting term S0 , which has quadratic form, the partition function path integral given in (5.19) was simply a generalized Gaussian Grassmann 62 integral of the kind in formula (5.11) and it could have been immediately written in a compact form. However, the interacting terms, Sc and S p , have quartic forms, therefore the Gaussian formula (5.10) cannot be used to compute the path integral. Fortunately, there is a standard way to get around this difficulty, introduced by Stratonovich [166] and Hubbard [86], hence its name: Hubbard-Stratonovich transformation. The idea is basically generalization of the following identity for higher dimensions 1 exp[− ax ] ≡ √ πa 2 Z y2 dy exp − − 2ixy , a (5.20) which through a shifted Gaussian integral over a new auxiliary variable y expresses exp[− ax2 ] in terms of an integrand in which x only appears linearly in the exponent. Similarly, to transform our path integral by Hubbard-Stratonovich (H-S) trick is to use another Gaussian path integral over new auxiliary fields to express exp[−Sc ] and exp[−S p ] in terms of an effective action, which is quadratic in ψ’s rather than quartic. Application of H-S for Sc gives us exp[−Sc ] = Zc−1 Z " D φ exp − ∑ m Vβ φm φm + iφm ∑ ψ̄l −m ψl 2Ueff l !# , (5.21) in which φ is a real field that is the corresponding auxiliary field for Coulomb interaction. √ The factor Zc−1 is a normalization constant similar to 1/ πa in (5.20). For the Cooper channel action, S p , it is convenient to introduce a complex auxiliary field. Then H-S transformation for exp[−S p ] would be exp[−S p ] = 1 Z− p Z " D ∆¯ D ∆ exp − ∑ n ∆n 4 Vβ ¯ ∆n ∆n + 2g ¯n ∆ ∑(eiθl−n/2 + eiθl+n/2 )ψ̄l+n/2 ψ̄−l+n/2 + 4 l (5.22) ∑(e−iθ − l n/2 l !# + e−iθl+n/2 )ψ−l +n/2 ψl +n/2 , in which ∆ is the complex auxiliary field that represents the emerging superconducting 1 −1 field. Here again, Z − p is a normalization factor same as Zc in (5.21). Now using (5.21) and (5.22), the full action, S = S0 + Sc + S p , acquires a quadratic form in terms of ψ̄ and ψ. Thus, the ψ̄-ψ part of the path integral becomes a Gaussian Grassmann integral of the form shown in (5.11) and therefore manageable. On this basis, we define an effective action, Seff , which will be obtained after performing the ψ̄-ψ part of 63 the path integral. After careful index manipulations and suitable rearrangements, all the terms quadratic in ψ̄ and ψ can be collected to express Seff as Z exp[−Seff ] = " # 1 ψl 0 D ψ̄D ψ exp − ∑[ψ̄l , ψ−l ] Mll 0 , ψ̄−l 0 2 ll 0 (5.23) with the matrix Mll 0 defined as − Gl−1 δll 0 + iφl 0 −l ∆l −l 0 (eiθl + eiθl 0 )/2 , Mll 0 = ¯ −1 0 0 ∆l 0 −l (e−iθl + e−iθl 0 )/2 G− l δll − iφl −l (5.24) in which Gl is the Green’s function defined in (5.16). Integrating ψ̄ and ψ out, we will be left by an action that only depends on the emerging bosonic fields φ and ∆. Note that in the expression (5.21) for Sc and (5.22) for S p , there are also terms with no ψ̄ or ψ in them. These terms form the noninteracting parts of the action for the φ and ∆ fields and we call (0) (0) them Sφ and S∆ , respectively. It should be clear from (5.21) and (5.22) that they are (0) Sφ (0) S∆ = Vβ ∑ 2Ueff φm φm , (5.25) Vβ ¯ ∆n ∆n . 2g (5.26) m = ∑ n Based on this effective action in terms of the emerging fields φ and ∆, we are going to study several aspects of our model in the sections that follow. 5.2 Zero temperature In this section, we focus on the zero temperature case. We will show how some of the results found in Chapter 3 and Chapter 4 can be derived in the path integral approach that we use here. Furthermore, we are going to show how the action framework allows us to study the fluctuations of the superconducting field. 5.2.1 Without Coulomb repulsion Following the pattern in Chapter 3 and Chapter 4, let us start with the simpler case where Coulomb repulsion is not present. This means that in the Matrix Mll 0 (defined in (5.24)), which is the core of the effective action Seff (given by (5.23)), we have to set φ = 0. Moreover, we are going to ignore fluctuations of the superconducting field too. That is to assume the superconducting field is a uniform unperturbed condensate, or in 64 ¯ n = ∆∗ δn,0 . These two assumptions together make mathematical terms, ∆n = ∆0 δn,0 and ∆ 0 Mll 0 a block diagonal matrix as − Gl−1 ∆0 eiθl Mll 0 = −1 δll 0 ≡ Ml δll 0 . ∆0∗ e−iθl G− l (5.27) Recall that according to the Gaussian Grassmann integral formula (5.11), to compute the path integral in (5.23) and find the effective action Seff , we need det M. Having a block diagonal matrix M makes this calculation a lot easier because for a block diagonal M, det M is given by the product of all det Ml . To compute det Ml , which is just a two by two determinant, one should use the expression (5.16) for Gl and remember that the time component of G−l has the opposite sign. Taking log from both sides of (5.23) to find Seff turns the product into a summation and gives us Seff ωn2 + ξ p2 + |∆0 |2 1 1 | ∆0 |2 = − ∑ ln = cte − ∑ ln 1 + 2 2 ωn ,p 4 2 ωn ,p ωn + ξ p2 ! , (5.28) where we are pulling out the constant part to form a converging integral that gives the contribution of the condensate, ∆0 , in the effective action Seff . Next realizing that the (0) contribution of the condensate in the noninteracting part of the action S∆ (given in (5.26)) is Vβ|∆0 |2 /2g, ignoring the constant part in (5.28), we can write the contribution of ∆ in the density of free energy as (0) f ∆0 = S∆0 + Seff (∆0 ) Vβ | ∆0 |2 1 | ∆0 |2 = − ln 1 + 2g 2Vβ ω∑ ωn2 + ξ p2 n ,p ! . (5.29) In the same way that in the limit of very large system size, V, the summation over momentum p becomes an integral, in the limit of very large β, that is as temperature goes to zero, summation over Matsubara frequencies becomes an integral. In fact, in that limit, we have the very similar association rule as 1 β ∑→ ωn 1 2π Z dω. (5.30) Using (5.30), the ωn summation in (5.29) can be done to obtain f ∆0 = | ∆0 |2 1 − 2g 2V ∑ p | ∆ |2 q0 . |ξ p | + ξ p2 + |∆0 |2 (5.31) 65 Now we argue that if f ∆0 has minimum at ∆0 6= 0, then reaching minimum free energy prefers non-zero ∆0 , hence the presence of superconducting state. Therefore, searching for the minimums, we take derivative of f ∆0 with respect to |∆0 | and get  ∂ f ∆0 1 1 = | ∆0 |  − ∂ | ∆0 | g 2V  ∑q p 1 ξ p2 + | ∆0 |2  = 0, (5.32) which always has a solution at ∆0 = 0. But since f ∆0 is increasing for large |∆0 |, if the bracket in (5.32) is zero, it would correspond to a minimum. Therefore, we conclude that the criterion for the system going into superconducting state is given by 1= g 2V ∑q p 1 ξ p2 + |∆0 |2 , (5.33) which is just the same as Eq. (4.2) we found in Chapter 4 based on BCS wavefunction. And as it is shown there, it can be approximated to get Eq. (4.3) that gives us the universal function |∆˜0 |(Λ̃) plotted in Figure 4.2, the function that describes different properties of the superconducting transition, including BEC-BCS crossover, as it’s discussed in the Section 4.1. 5.2.2 Effects of Coulomb repulsion Now we are going to add Coulomb repulsion while still ignoring fluctuations of the ∆ field. In this case, due to the presence of φll 0 , the matrix M is no longer block diagonal. As a result, its determinant cannot be computed in a simple, straightforward fashion. Therefore, we need to follow an approximation scheme. First, note that due to the matrix identity (see for example Theorem 2.12 in Ref. [81]) exp[Tr ln M ] = det M, (5.34) the path integral in (5.23) can be equivalently written as 1 exp[−Seff ] = exp[ Tr ln M ], 2 (5.35) in which the factor 1/2 is due to the fact that in the path integral (5.23), unlike the generic Gaussian Grassmann integral formula (5.11), both ψ and ψ̄’s are present on the both sides 66 of the matrix M (see Ref. [95] for more details). Next for developing an expansion scheme, the following factorization of M is useful − Gl−1 δll 0 0 iGl φl 0 −l M= 1 − −1 ∗ −iθl δ 0 0 − G 0 G− δ ll − l ∆0 e l ll Gl ∆0 eiθl δll 0 iG−l φl −l 0 Then combining (5.35) and (5.36), we can say 1 − Gl−1 δll 0 0 iGl φl 0 −l Seff = − Tr ln + ln 1 − −1 − G−l ∆0∗ e−iθl δll 0 0 G−l δll 0 2 . (5.36) Gl ∆0 eiθl δll 0 iG−l φl −l 0 , (5.37) in which the first term amounts to a constant, equivalent to the constant in (5.28), that does not depend on either ∆ or φ and hence can be ignored. Finally, utilizing the series, ∞ ln(1 − x ) = − ∑ n =1 xn , n (5.38) we shall write the formal expansion series for the effective action Seff as n ∞ 1 1 iGl φl 0 −l Gl ∆0 eiθl δll 0 Seff (∆0 , φ) = Tr ∑ . 2 n=1 n − G−l ∆0∗ e−iθl δll 0 iG−l φl −l 0 (5.39) The first thing that is rather straightforward to check about the series (5.39) is that in the absence of φ, trace of all the odd terms vanishes. Then, thanks to the presence of δll 0 in the off-diagonal ∆ terms, trace of the even terms simplify to a sum over just one index to give us ∞ 1 1 0 Seff (∆0 ) = Tr ∑ 2 n=1 n − G−l ∆0∗ e−iθl δll 0 Gl ∆0 eiθl δll 0 0 n − G−l Gl |∆0 |2 1 = ∑∑ 2 l n n n , (5.40) which, after using the series identity (5.38) and (5.16) for the Green’s functions, reproduces what we found in (5.28). (0) The noninteracting part of the action for the condensate, S∆0 = Vβ|∆0 |2 /2g, is a parabola and has a minimum at |∆0 | = 0. The presence of attractive interaction modifies this shape and brings the possibility of having a minimum at some non-zero |∆0 |. If there is such a minimum, then one can conclude that the superconducting state is stable. The criterion for having such a non-zero minimum can be developed by expanding the action for small |∆0 |. Then the sign of the coefficient of the quadratic term, |∆0 |2 , would tell us about the existence of a minimum at |∆0 | 6= 0; if the coefficient is positive, the minimum is still at |∆0 | = 0, and when it becomes negative, we start getting a non-zero minimum. Therefore, to find the critical point, after expanding (5.39), one should collect the coefficient 67 of |∆0 |2 and set it equal to zero. Following this strategy, next we present the contribution of the few first terms in the expansion (5.39) to the coefficient of the term |∆0 |2 . The first term in the expansion that generates |∆0 |2 is, of course, the second term (n = 2). With careful bookkeeping, it can be seen that the contribution of this term to the coefficient of |∆0 |2 is given by (2) Seff,∆2 = − 0 | ∆0 |2 2 ∑ Gl G−l . (5.41) l The third term (n = 3) also generates terms with |∆0 |2 . However, such terms would have a linear presence of φ in them. In the approximation scheme that we follow here, we are going to replace φ’s by their average value using only the noninteracting part of (0) (0) the φ-action, Sφ that is given in (5.25). But since Sφ is even with respect to all φm ’s, more (0) generally speaking, the average of the product of any odd number of φ’s over exp[−Sφ ] is going to be zero. Therefore, in our approximation, the contribution of the third term (and similarly all higher order odd terms) to the coefficient of |∆0 |2 is zero. The next term in the expansion that gives a non-zero contribution to the coefficient of |∆0 |2 is the fourth term (n = 4). After carefully collecting the relevant terms, that contribution can be written as (4) Seff,∆2 = 0 | ∆0 |2 2 + ∑0 Gl G−l (Gl Gl0 φl0 −l φl−l0 + G−l G−l0 φl−l0 φl0 −l ) | ∆0 2 ll |2 ∑0 Gl G−l Gl0 G−l0 φl−l0 φl−l0 eiθ −iθ 0 , l l (5.42) ll in which the first term would amount to zero in our approximation. That’s because the (0) average of a pair of φ’s over Sφ is only non-zero if their indices match. In fact for a pair of φ’s, we have hφm φm0 i = Zc−1 Z " # Vβ U φm φm0 exp − ∑ φm φm D φ = eff δm,m0 , 2U Vβ eff m (5.43) and we are going to replace φ pairs by their average value given above. Next, we are going to assume zero temperature and use the rule (5.30) to perform the ω parts of the integrals in (5.41) and (5.42). To do that, the following integral (that can be easily done using residue theorem) is used 1 2π Z dω sgn(ξ ) = . (iω − ξ )(−iω − ξ ) 2ξ (5.44) 68 In order to make our point, let us consider the cases where chemical potential is below the band (Λ > 0). In those cases, ξ p is positive for all p’s and so we don’t have to worry about the sign function. With this assumption and using (5.44) to perform the ω-integrals in (5.41) and (5.42), we can finally write (0) (2) (4) S∆0 + Seff,∆2 + Seff,∆2 0 Vβ 0 | ∆0 |2 = 2 " 1 1 − g 2V ∑ p 1 1 + ξ p 4V 2 ∑ p1 p2 # Ũ p1 ,p2 , ξ p1 ξ p2 (5.45) where we are using our previous notation, Ũ p1 ,p2 = U p1 − p2 (1 + cos(θ p1 − θ p2 ))/2. The bracket gives the first few terms in the expansion of the |∆0 |2 coefficient. Therefore, setting it equal to zero gives the first few terms in the expansion of the critical g. However, this expansion (at least to this order) only gives a reliable result for gc if the Coulomb is wellscreened and short-range. One may want to conclude that the method at hand doesn’t provide a reasonable gain to pain ratio at least as long as treating a long-range Coulomb repulsion is concerned. For that from now on we are going only to consider the wellscreened Coulomb and use the action method for the purposes that it is good for, namely the finite temperature analysis and studying fluctuations effects. In this and the previous subsections, we showed how the path integral approach could reproduce some of the results that we found in Chapter 3 and Chapter 4. Next, we are going to show how it provides a suitable framework to study some other aspects of the problem such as the effects of fluctuations. 5.2.3 Fluctuations of the superconducting field So far we only considered the condensate part of the superconducting field |∆0 |. That was manifested mathematically by assuming ∆l −l 0 = ∆0 δll 0 . Now we are going to relax that constraint and instead ignore the Coulomb field again. One can think of fluctuations as collective excitations on top of the uniform superconducting field. Our goal is to find the dispersion relation of such excitations for long wavelength and draw a couple of conclusions from it. To do that, we go along the following strategy. First, we find an approximate action that describes the condensate part of the superconducting field. That will allow us to approximate the magnitude of the condensate |∆0 |. Then we are going to find an approximate fluctuation action, sitting on top of the condensate field, that describes long wavelength fluctuations. Moreover, since we are working at zero temperature, we 69 are going only to consider cases where the chemical potential is below the band and ξ p is positive for all p’s. That is because in those cases, some finite attraction g is required to get a superconducting state. That allows for having infinitesimally small condensate, |∆0 |, at zero temperature that justifies expanding the action for the ∆-field. In order to form the approximate action for the condensate, we use the noninteracting (0) part of the action, S∆0 given in (5.26), and the first two terms of the expansion (5.40) to write (2) (4) S∆0 0 + Seff,∆0 + Seff,∆0 Vβ | ∆0 |2 = 2 " 1 1 − g Vβ ∑ G−l Gl # + l | ∆0 |4 1 2 2 G− l Gl , 4 Vβ ∑ l (5.46) which has the following Ginzburg-Landau form (2) (4) S∆0 0 + Seff,∆0 + Seff,∆0 Vβ = α0 | ∆0 |2 | ∆0 |4 +γ , 2 4 (5.47) where the coefficients α0 and γ are given by the following Green’s function integrals that are rather straightforward to compute. For the ω part of the integrals, we use the zero temperature rule (5.30) and for the p-integral, we use pdp ∼ p0 dp and extend the integral to infinity as usual. α0 = γ = 1 1 − g Vβ 1 Vβ ∑ G−l Gl = l 1 1 − g V 1 1 ∑ G−2 l Gl2 = V ∑ 4ξ 3 l p p 1 ∑ 2ξ p = = p 1 1 − g gc0 3 , 16gc0 Λ2 (5.48) (5.49) where gc0 is the critical attraction in the absence of Coulomb repulsion and is given by (3.5). At g = gc0 , the G-L action (5.47) acquires a non-zero minimum given by |∆0 |2 = −α0 /γ where the minimum value of the action itself is −α20 /4γ. Notice how the minimum free energy for the condensate action only determines the magnitude of the condensate |∆0 |. However, the superconducting field ∆ is a complex field. That means that the norm of the condensate ∆0 is determined by minimizing the free energy, but it can have any phase. In reality, it would spontaneously pick a phase in the same way that ferromagnets spontaneously pick a magnetization direction. This phenomenon is called spontaneous symmetry breaking. Now since we are free to pick any phase for the condensate, we choose its phase to be zero; that is to assume ∆0 is a positive real number. Now our task is to see how the long wavelength fluctuations play on top of the condensate. We are going to construct an effective fluctuation action that has quadratic form in 70 ¯ n and ∆n – the noncondensate part of the ∆-field. The noninteracting part, S(0) terms of ∆ ∆ given in (5.26), already has a simple quadratic form. In order to collect the higher order terms, we shall apply the same factorization and expansion scheme in (5.36) – (5.39) on the original effective action (5.24) in the absence of the Coulomb repulsion. That gives us the following expansion for the effective action in which fluctuations are included. m ∞ 1 1 0 Gl ∆l −l 0 (eiθl + eiθl 0 ) . Seff (∆) = Tr ∑ m ¯ l −l 0 (e−iθl + e−iθl 0 ) 0 2 m=1 2 m − G−l ∆ (5.50) The first non-zero term in (5.50) is the second term that is after proper index manipulation, and simple algebra can be written as (2) Seff,∆ = −∑ n ¯ n ∆n ∆ 2 ∑ Gl Gn−l cos 2 l θl − θl −n 2 , (5.51) ¯ n ∆n . Next, we in which the summation over l gives the coefficient of the quadratic form ∆ are going to approximate this coefficient for the long wavelength limit. Using the zero temperature rule (5.30) and residue theorem, the ω part of the l-summation in (5.51) can be done to obtain ∑ Gl Gn−l cos2 l θl − θl −n 2 = β∑ p cos2 θ p − θ p−Q 2 ξ p + ξ Q−p − iωn , (5.52) which is a function of the four-vector index n = (iωn , Q) that belongs to the fluctuations of the superconducting field. Since we are interested in the long wavelength limit (recalling that the Rashba momentum, p0 , determines our length scale) we expand the integrand in (5.52) for Q p0 and ωn ξ p as cos2 θ p − θ p−Q 2 ξ p + ξ Q−p − iωn ∼ 2 1 iωn κ ( p − p0 ) Q cos θ 2ER (2Λ − ξ p ) + ξ p Q2 sin2 θ + 2 + − , (5.53) 2ξ p 4ξ p 2ξ p2 8ξ p3 p20 where θ is the angle between p and Q and following our previous notation, ER = κ p20 , is the Rashba energy. Next we use the expansion (5.53) to calculate the integral in (5.52). Notice that the linear term in Q vanishes over the angular integral. Then the p integral is done by approximating pdp ∼ p0 dp and extending it to infinity to finally obtain ∑ Gl Gn−l cos l 2 θl − θl −n 2 Vβ iωn κQ2 ∼ , 1+ − gc0 4Λ 16Λ (5.54) 71 where we also used Λ ER to approximate the result. gc0 here as it was introduced in (3.5) is the critical g in the absence of Coulomb repulsion. Now using (5.54) in (5.51) the contribution of the second term for long wavelength fluctuations can be written as (2) Sfluc,∆ Vβ =− 2gc0 iωn κQ2 ¯ ∑ ∆n ∆n 1 + 4Λ − 16Λ . n (5.55) The next non-zero term in (5.50) is the fourth term. However, as it is expected, in the ¯ l −l ∆l −l ∆ ¯ fourth term of (5.50), ∆ appears in quartic form; it contains products like ∆l1 −l2 ∆ . 2 3 3 4 l4 − l1 In order to make an effective quadratic form out it, we are going to set two of the four at the condensate, that is to set them equal to ∆0 δll 0 . There are six different ways to choose ¯ one has ∆∆ form, and one is two out of four. Four out of the six ways have the form ∆∆, ¯ ∆. ¯ The last two terms are usually called anomalous terms. Following in the form of ∆ this approximation scheme, the effective quadratic contribution of the fourth term into the fluctuation action can be written as (4) Sfluc,∆ =∆20 +∆20 ∑ ∆¯ n ∆n ∑ Gl2 G−l Gn−l cos2 n l ∑ n ¯ n∆ ¯ −n + ∆n ∆−n ∆ 4 θl − θl −n 2 ∑ Gl G−l Gl−n Gn−l cos 2 l θl − θl −n 2 , (5.56) where the anomalous terms are written in the second line. Notice that for small n = (iωn , Q), both coefficients in (5.56) are proportional to ∆20 (γ + corrections) where γ is given in (5.49). But since ∆20 itself is assumed to be infinitesimally small, we can neglect the corrections and simply write (4) Sfluc,∆ = − Vβα0 4 where we used ∆20 = −α0 /γ. ∑ 4∆¯ n ∆n + ∆¯ n ∆¯ −n + ∆n ∆−n , (5.57) n (0) (2) (4) Finally collecting all three terms, S∆ in (5.26), Sfluc,∆ in (5.55), and Sfluc,∆ in (5.57), we can write our effective fluctuation action in the following quadratic form Sfluc Vβ = 16Λgc0 ∑ n ¯n ∆ ∆−n T κ with Q0 defined as Q20 4 (Q 2 + Q20 ) − iωn κ 2 4 Q0 Λ 16 = −16α0 gc0 = 2 κ ζ where ζ is the pair size defined in (3.6). κ 2 4 Q0 κ 2 2 4 ( Q + Q0 ) + iωn gc0 1− g ∆n ¯ −n , ∆ (5.58) , (5.59) 72 Now that we have the effective quadratic fluctuation action, we can study a few things about them. First of all, notice how we had to introduce off-diagonal elements in (5.58) to accommodate the anomalous terms. In the absence of the off-diagonal elements, the dispersion relation of each mode is simply given by the diagonal. Therefore, to find the dispersion relation of the fluctuations, one has to diagonalize (5.58). This can be done by Bogoliubov transformation. Alternatively, one can just look for the zeros of the action determinant as they don’t change under Bogoliubov transformation. Following this simpler alternative, we can write κ 2 κ 2 κ 2 2 2 ( Q + Q20 ) − iωn Q0 4 4 = Q ( Q + 2Q20 ) + ωn2 det κ 2 κ 2 2 Q ( Q + Q ) + iω 4 n 0 4 0 4 = κ Q 4 q Q2 + 2Q20 − iωn κ Q 4 q Q2 + 2Q20 + iωn = (ωQ − iωn )(ωQ + iωn ), (5.60) that gives us two modes where the second one is just the time-reversal copy of the first one. These modes have two important features. First, they are gapless; that is to say, it takes infinitesimally small energy to excite an infinitely long wavelength excitation on top of the superconducting field. The existence of this gapless mode is the consequence of the spontaneous symmetry breaking. Yoichiro Nambu was the first who discovered the presence of these gapless modes in superconducting fields [121]. Shortly after him, Jeffrey Goldstone shed more light on the phenomenon [77] and later, in collaboration with Salam and Weinberg, generalized it to any system where some continuous symmetry spontaneously breaks down [78]. Such a collective mode is now known as the Goldstone mode of the system. The second point is that the modes are not only gapless but are also linear in Q for Q → 0, similar to acoustic modes of lattice vibrations – which are in fact the Goldstone mode of crystals as they break the continuous translational symmetry of the space. It is easy to see that for Q Q0 , we have κQ0 p0 gc0 ωQ ∼ √ Q = √ 2 2 2 2 gc0 1− g Q, (5.61) which means that the more g goes beyond the critical gc0 , the faster the speed of acoustic waves becomes. Moreover, this tells us that the acoustic speed is also proportional to p0 gc0 . The fluctuation action also enables us to calculate the total power of fluctuations. Comparing the fluctuation power with the condensate, we can evaluate how robust it is against 73 the fluctuation. The total fluctuation power can be defined as h(δ∆)2 i = ∑h∆¯ n ∆n i, (5.62) n ¯ n ∆n i. Using the fluctuation action (5.58), correspondwhich requires us to first calculate h∆ ing partition function can be made to compute the required average. The average is given by the following path integral −1 h∆¯ n ∆n i = Zfluc Z κ ( Q2 + Q20 ) + iωn ¯ n ∆n exp[−Sfluc ]D ∆ ¯ D ∆ = 8Λgc0 4 ∆ , 2 + ω2 Vβ ωQ n in which Sfluc is given by (5.58) and Zfluc = R (5.63) D ∆¯ D ∆e−Sfluc . Note that in the path inte- ¯ ±n and ∆±n cancel with their corresponding gral (5.63), all terms except those involving ∆ ¯ ±n and ∆±n is a standard Gaussian-related integral terms in Zfluc , and the part involving ∆ that gives us the above result. ωQ in (5.63) is the dispersion relation of the fluctuations defined in (5.60). Now using (5.63) in (5.62), we can write the total fluctuation power as h(δ∆)2 i = 8Λgc0 Vβ ∑ ωn ,Q κ 2 4 (Q + Q20 ) + iωn . 2 + ω2 ωQ n (5.64) In performing the ω-part of the integral in (5.64) one should note that the imaginary part of it is not-strictly converging though it seems to be zero for being odd with respect to ωn . In fact, having nonconverging tail for large ω plus being odd makes it possible to draw different values from it using different limits. Here we use the following prescription for the physical reason that comes ahead 1 lim + η →0 2π Z dω iωeiωη 1 =− . 2 2 2 ωQ + ω (5.65) Using (5.65) prescription and carrying out ω-part of the integral in (5.64), we obtain h(δ∆)2 i = 4Λgc0 V ∑ Q Q2 + Q20 q − 1, Q Q2 + 2Q20 (5.66) where the −1 comes from the imaginary part carried out using (5.65). With the −1, the integrand of the Q-integral in (5.66) goes to zero as Q0 → 0. But Q0 goes to zero as the transition occurs and superconductivity disappears and when there is no superconductivity, there cannot be fluctuations of the superconducting field either. And this is the physical reason behind choosing (5.65) prescription; that is to get a sensible result at the point of transition. Moreover, thanks to the presence of −1 in its integrand, the Q-integral 74 in (5.66) is convergent for Q → ∞ as well. Therefore, we can extend it to infinity to finally get 2Λgc0 Q20 h(δ∆) i = π 2 Z ∞ 2 x +1 0 √ x2 + 2 − x dx = Λgc0 Q20 . π (5.67) Now we are ready to compare h(δ∆)2 i in (5.67) with ∆20 = −α0 /γ. Utilizing (5.59) for Q20 and (5.49) for γ and after a little bit of algebra we can write s h(δ∆)2 i < ∆20 → Λ π < , ER 12 (5.68) which is consistent with the condition we are working in, i.e., Λ ER . From this, one can conclude that although our system is two dimensional, at zero temperature the superconducting field is stable against its own fluctuations. In the next section, analyzing fluctuations at finite temperature, we show that finite temperature combined with low dimensionality of the system can disrupt this stability. 5.3 Finite temperature As we said at the beginning of this chapter, one of the advantages of path integral approach is that by constructing partition function of the system, it provides a systematic way to do every calculation at finite temperature. This is essentially done by abandoning the zero temperature rule (5.30) and appreciating the discreteness of the summations over ωn ’s, the Matsubara frequencies (see (5.6) and (5.8)). Handling ωn -summations is typically more challenging than the zero temperature ω-integrals (that are usually easy to compute using residue theorem). Moreover, on the conceptual side, calculating at finite temperature is to add one more degree of freedom, the temperature, that is also a new energy scale. Combination of all these computational and conceptual challenges makes the finite temperature analysis more complicated. Here in this section to focus on the effects of finite temperature, we only consider the superconducting field, ∆, and ignore the Coulomb part of the Hamiltonian (or equivalently the φ field in the effective action (5.24)). This can be justified by assuming a strongly screened Coulomb repulsion that – as we showed in Chapter 3 and Chapter 4 – only manifests itself in replacing the attraction g by some effective attraction geff . With that brief introduction, next we start by the simplest case, that is to neglect the fluctuations as well. 75 5.3.1 Without fluctuations As we discussed before, the condensate part of the action can be approximated by a G-L action already built in (5.47) with α0 and γ coefficients defined in (5.48) and (5.49), respectively. And we saw that the existence of a superconducting condensate is determined by the sign of α0 . Therefore, to find the critical temperature of our system, all we need to do is to calculate α0 at finite temperature. To do that, we need to compute the following summation 1 β 1 1 ∑ Gl G−l = β ∑ ω2 + ξ 2 , ωn ωn n (5.69) p where ωn ’s are fermionic Matsubara frequencies as they belong to electrons. The standard trick to compute this class of summations is to notice that for fermionic Matsubara frequencies, iωn ’s are the zeros of exp[ βz] + 1. Therefore, including (exp[ βz] + 1)−1 in the integrand of a closed loop complex integral that goes around the whole complex plane can generate a summation over ωn ’s because iωn ’s are going to be its poles. For example, in our specific case, we can write I " tanh( βξ p /2) dz 1 = 2πi − + βz 2 2 2ξ p β (e + 1)(z − ξ p ) ∑ ωn 1 2 ωn + ξ p2 # = 0, (5.70) in which the first term in the bracket comes from the residues at the poles of (z2 − ξ p2 )−1 . The loop integral in (5.70) goes to zero because its integrand decays faster than R−2 if R → ∞ is the radius of the loop that goes around the whole complex plane. From (5.70), we can deduce the result of the required summation in (5.69). Computing the discrete ωn -summation of α0 in the above way, now it can be written in terms of a p-integral as follows α0 = 1 1 − g V ∑ p tanh( βξ p /2) . 2ξ p (5.71) Setting α0 equal to zero gives us the the critical temperature, Tc , for any given g and Λ, where Λ now could be either positive or negative, pertaining to chemical potential being below or inside the moat band, respectively, while we are still working in the limit |Λ| ER . Then in the same way that we derived Eq. (4.3) for ∆ and Λ in Chapter 4, we can approximate the integral in (5.71) and normalize every energy to Λc0 = g2 p40 /16ER to get an implicit equation for Tc and Λ as 1= 1 π Z ∞ −∞ dx x2 + Λ̃ tanh , 2T̃c x2 + Λ̃ (5.72) 76 where Λ̃ = Λ/Λc0 and T̃c = Tc /Λc0 . Equation (5.72) implicitly defines a universal function T̃c (Λ̃) that gives the relation between Tc and Λ for any g upon a proper scaling. Solving Eq. (5.72) numerically, the universal function can be plotted as in Figure 5.1. Comparing it ˜ (Λ̃), one would admit with Figure 4.2 that shows the zero temperature universal function ∆ that they are very much similar to each other. That is of course expected as generally, the critical temperature follows the magnitude of the order parameter at zero temperature. Mathematically arguing, this is because the integrands in both Eq. (5.72) and Eq. (4.3) are peaks with roughly the same shapes: same hights, same widths, and same tails. In fact, one can check that we have 1 ξ2 tanh[ξ/t] ∼ − 3, ξ t 3t 1 1 ξ2 for |ξ | 1 : p ∼ − 3, d 2d ξ 2 + d2 for |ξ | 1 : tanh[ξ/t] 1 ∼ , ξ |ξ | 1 1 for |ξ | 1 : p ∼ . |ξ | ξ 2 + d2 for |ξ | 1 : (5.73) (5.74) The BEC-BCS crossover is visible in the behavior of the critical temperature too. In the vicinity of the band edge, where |Λ̃| ∼ 1, the critical temperature is of the order of Λc0 that itself is proportional to g2 . This is the signature of the BEC regime, where the superconducting state is a Bose-Einstein Condensate of well-defined molecular-like electron pairs. On the BCS side, one can show from Eq. (5.72) that for very large negative Λ̃, p we have Tc ∝ |Λ| exp[− π2 |Λ|/Λc0 ] that has the familiar form of exp(− A/g) for critical temperature in the standard BCS theory. Moreover, from Figure 5.1 it is visible that the maximum critical temperature happens at the negative side of Λ, as we had expected from zero temperature calculation. Although comparing the location of the maximum point (the point B) in Figure 5.1 and Figure 4.2 one would realize that there is a small difference between Λ̃m,Tc and Λ̃m . The more dra˜ (Λ̃) however, is in their asymptotic behavior at the matic difference between T̃c (Λ̃) and ∆ ˜ near transition point, Λ̃ = 1. As we mentioned in Section 4.1, asymptotic behavior of ∆ q ˜ ∼ 8 (1 − Λ̃), which actually turns out to be very different from the transition point is ∆ 3 T̃c asymptote. Using Eq. (5.72), the asymptotic behavior of T̃c in the close vicinity of the transition point can be found by 1 − Λ̃ = 4 s T̃c 1 2 , exp − → T̃c = π T̃c W π (132 2 −Λ̃) (5.75) 77 1.0 B 0.8 A T̃c 0.6 0.4 0.2 0.0 -15 -10 -5 Λ̃m,Tc 0 1 Λ̃ Figure 5.1. The universal function T̃c (Λ̃). Two important points of the function are shown in the figure: A is when the chemical potential touches the band edge where [Λ̃0 , T̃c,0 ] ≈ [0, 0.736] and B is the maximum Tc point where [Λ̃m,Tc , T̃c,m ] ≈ [−1.784, 0.888] where W is the positive branch of the Lambert-W function (see Chapter 4.13 in Ref. [50]). The asymptotic behavior (5.75) indicates that at the transition point, Tc grows qualitatively much faster than the order parameter ∆. That is because the rhs of the first equation in (5.75) exponentially goes to zero as T̃c goes to zero. Taking this point about the asymptotes at the transition point as the last noteworthy feature of the superconducting condensate at finite temperature, next, we are going to see how the fluctuations play on top of the condensate when the temperature is not zero. 5.3.2 Fluctuations of the superconducting field As we said, one of the beauties of path integral approach is that it produces a formulation that can be used both for zero and finite temperature analysis. Therefore, all we need to do for studying fluctuations at finite temperature is to follow every step we 78 took in Section 5.2.3 but do the calculations at finite temperature; that is to appreciate the discreteness of ωn summations. From zero temperature calculations, recall that the fluctuation action is built out of three (0) pieces. First, there is a noninteracting part, S∆ given in (5.26), that involves no integration or summation and so will stay the same at finite temperature. The next term comes from (2) the second-order term in Seff series (5.50), Seff,∆ given in (5.51). This term will be affected by finite temperature as it involves a summation over l = (iωn0 , p). In fact, in a way, this is the most important term as it is the only where the long wavelength fluctuations, n = (iωn , Q), manifest themselves. The last term is the effective quadratic contribution of the fourth term in the series (5.50), given in (5.57). The finite temperature effects on this term are hidden in its prefactor α0 that we already have taken into account in (5.71). (2) Therefore, all we need to do here is to calculate Seff,∆ in (5.51) at finite temperature. As the first step using a similar trick introduced in (5.70), we find n F (ξ p ) − n F (−ξ Q−p ) 1 =β , 0 −ξ ) − ξ )( iω − iω iωn − ξ p − ξ Q−p p n Q−p n n ∑0 Gl Gn−l = ∑0 (iω0 ωn ωn (5.76) where n F (ξ ) = (exp[ βξ ] + 1)−1 is the famous Fermi distribution. Then in the spirit of (5.48), we can define αn = = 1 1 − g Vβ 1 1 − g V ∑ Gl Gn−l cos2 l ∑ p θl − θl −n 2 n F (ξ p ) − n F (−ξ Q−p ) cos2 iωn − ξ p − ξ Q−p θ p − θ p−Q 2 . (5.77) In order to approximate αn for small n = (iωn , Q), we approximate α(0, Q) and α(iωn , 0) separately and then put the corrections coming from them together. That can be done analytically only in two limits: first in the BCS limit when the chemical potential is in the band and Tc |Λ|, and second close to the critical point when the chemical potential is below the band and Λ ∼ Λc0 ; the latter is when the BEC of superconducting pairs is emerging and we again have Tc Λ (see Figure 5.1). Let us start calculation of the BCS case by approximating α(0, Q) − α0 . It is convenient to use the shift p → p + Q/2 in (5.77). We also use the following expansions ξ p±Q/2 ∼ ξ p ± κ ( p − p0 ) Q cos θ ≡ ξ p ± e, Q2 2 θp+Q/2 − θp−Q/2 cos ∼ 1 − 2 sin2 θ, 2 4p0 (5.78) 79 in which θ is the angle between p and Q. Using the above shift and expansions, we can write α(0, Q) − α0 ∼ 1 2V tanh( βξ p /2) sinh( βξ p ) − ∑ ξp ξ p cosh( βξ p ) + cosh( βe) p 1− Q2 2 sin θ . 4p20 (5.79) This includes two correction terms; first is the normal one that we had if the phase factor cos2 was not present, and the second is coming from the correction Q2 sin2 θ/2p20 that the phase factor introduces. For the first term extending the integral around p0 as usual and using proper rescaling, we get the following integral Z Z p20 dx β|Λ| 2 sinh( β|Λ|( x2 − 1)) p dθ tanh ( x − 1) − . x2 − 1 2 cosh( β|Λ|( x2 − 1)) + cosh( βe) 8π 2 ER |Λ| (5.80) To approximate this integral, we use the following integral approximation that for | a| 1 Z sinh(z) a a2 tanh(z/2) − ∼ tanh2 ∼ , (5.81) dz z z(cosh(z) + cosh( a)) 2 4 in which the integrand is a peak centered at z = 0 with a width of order of 1, a hight that is proportional to a2 , and an exponentially converging tail. Comparing (5.80) and (5.81), one should realize that the x part of the integral (5.80) can be approximated by two integrals of the form (5.81) that are centered around x = ±1. Doing so, the first correction term becomes sinh( βξ p ) tanh( βξ p /2) − ξp ξ p cosh( βξ p ) + cosh( βe) p p Z p20 ER | Λ | 2 β2 κ |Λ| Q2 cos2 θ p (5.82) ∼ dθ = Q , 2 2 4 32πT 8π ER | Λ | p where we also used e = κ ( p − p0 ) Q cos θ ∼ ± κ |Λ| Q cos θ, that is to replace p − p0 by its 1 2V ∑ value at the Fermi points. Also, it’s worth mentioning that our assumption βe 1 implies p that this expansion is valid only when Q/p0 T/ ER |Λ| 1. For the second term since it already has a factor of Q2 /p20 , we can simply approximate cosh( βe) ∼ 1 to get 1 Q2 V 4p20 tanh( βξ p /2) 1 sin2 θ = ∑ 2ξ p 2 p 1 − α0 g Q2 , 4p20 (5.83) where we used the fact that the integral is exactly the same as the one that gives α0 in (5.71) except that it has an angle dependence sin2 θ that generates the factor of 1/2. Finally, since 80 we are working just below the critical temperature, we shall set T = Tc in (5.82) and α0 = 0 in (5.83) and put them together to obtain p α(0, Q) − α0 = ER |Λ| p20 1 + 2 4πTc g ! Q2 , 8p20 (5.84) in which the temperature-dependent term is the normal term and in the limit that we are working in, i.e., |Λ| Λc0 , is larger than the term coming from the phase factor that goes as 1/g. Staying in the BCS limit, |Λ| Λc0 , now we are going to approximate α(iωn , 0) − α0 . Using (5.71) and (5.77) and after a simple algebra, we have 1 α(iωn , 0) − α0 = V ∑ tanh p βξ p 2 iωn ωn2 − 2ξ p (4ξ p2 + ωn2 ) 4ξ p2 + ωn2 ! , (5.85) where we separated the real and imaginary parts of the integrand. The main contribution of this integral comes from the points where ξ p = 0, which are of course, in the BCS limit that we are considering, at the Fermi momenta or the inner and outer edges of the moat ring. But the imaginary part of integrand is an odd function of ξ p around the Fermi points and therefore, its contribution is negligible. Keeping only the real part, performing the angular integral and extending the integral around p0 as usual, we get p2 p0 α(iωn , 0) − α0 = 4π ER |Λ| Z dx tanh( β|Λ|( x2 − 1)/2) ωn2 , x2 − 1 4|Λ|2 ( x2 − 1)2 + ωn2 (5.86) in which the integrand is basically a product of two narrow double peaks centered at x = ±1. The first peak has a height of β|Λ|/2 1 and a width of 1/β|Λ| 1. The height of the second one is 1 and its width is |ωn |/|Λ| 1. Therefore, the height of the product is also β|Λ|/2 and its width is determined by the narrower one, and since we are working in the limit that ωn → 0, the second one determines the product’s width. Therefore, considering that there are two of these peaks, we can estimate the integral in (5.86) by twice of the height of the first peak times the width of the second one and write α(iωn , 0) − α0 ∼ p2 | ωn | p0 , 4π ER |Λ| Tc (5.87) where recalling that we are working just below the critical temperature, we also set 1/β = T → Tc . The fact that α(iωn , 0) − α0 goes as |ωn | instead of −iωn is the most important feature of the fluctuations in the BCS limit. To see that it’s enough to neglect α0 as we are 81 working in the close vicinity of the critical temperature. Setting α0 equal to zero makes the off-diagonal elements of the fluctuation action zero as well. Then the dispersion relation of the fluctuations is simply given by the diagonal term upon replacing iωn by ω, which is equivalent to going from the imaginary time, τ, to the real time, t. Using (5.84) and (5.87) and going to real time, we shall find that the dispersion relation of the fluctuations in the BCS limit is given by ωQ = i ! p ER |Λ| 4πTc ER |Λ| Q2 + , Tc gp20 8p20 (5.88) which means that any long wavelength fluctuation would die out in time because they evolve as exp(iωQ t). This is because in the BCS limit, the superconducting fluctuations couple to the conducting electrons at the Fermi level and dissipate their energy. However, one should note that we performed our calculation right at the transition point where the superconducting gap is not developed to protect pairs from mixing with the Fermi sea electrons. In fact, it can be shown that well below the transition temperature, the fluctuations do propagate in the superconducting gap. In such a calculation, one will q need to use the gapped dispersion of the electrons and holes, ± ξ 2p + ∆2 , instead of the free electron dispersion, ξ p . A proper calculation in that limit for a conventional metallic superconductor can be found in [133]. The other extreme limit at which we can study the behavior of superconducting fluctuations is in the vicinity of the critical point, Λ = Λc0 , where the BEC of electron pairs just starts to form and when we have Tc Λ. We will see that in this limit, the superconducting fluctuations are not suppressed and exhibit very different behavior. In this limit since βΛ 1, the occupation factor with exponential accuracy is 1; this allows us to use the expansion (5.53) and write α(iωn , Q) − α0 ∼ ∼ 2 iωn 2ER (2Λ − ξ p ) + ξ p Q2 sin2 θ ∑ − ξ2 + 2ξ p3 p20 p p p2 κ √0 −iωn + Q2 , 4 16Λ ER Λ 1 4V ! (5.89) where in performing the integral, we followed the usual scheme of extending it around p0 and we also used the fact that Λ ER . Next, recalling that the fluctuation action can be written in terms αn and α0 , using the above long wavelength expansion in (5.89), we can write it as 82 Sfluc ¯n ∆ ∆−n T αn − 2α0 − α0 ∆n ∑ ¯ −n − α0 α−n − 2α0 ∆ n κ 2 ¯ n T κ ( Q2 + Q2 ) − iωn p20 Vβ Q0 ∆n ∆ 0 4 4 √ = ∑ κ 2 κ 2 2 ¯ −n , ∆ 64Λ ER Λ n ∆−n 4 Q0 4 ( Q + Q0 ) + iωn Vβ = 4 (5.90) where Q0 is proportional to the condensate and is given by Q20 = −64α0 3∆20 p20 Λp ER Λ = . ER ΛER (5.91) Comparing (5.90) and the zero temperature fluctuation action in (5.58), one should realize that they are formally the same, except that here at finite temperature calculation, we fix the attraction g and α0 goes to the negative side as soon as the temperature goes below the critical temperature Tc . Clearly here in this limit, the fluctuation modes are not damped and moreover, the formal similarity immediately implies that their dispersion relation is given by a similar acoustic form in (5.60) as ωQ = κ Q 4 q Q2 + 2Q20 . (5.92) Now we wonder if these long wavelength fluctuations can overcome the condensate energy. Following what we did at zero temperature calculation in (5.63) and (5.64), here we can write the total power of the superconducting fluctuations as √ κ 2 2 32Λ ER Λ 2 4 ( Q + Q0 ) + iωn . h(δ∆) i = ∑ 2 + ω2 p20 Vβ ωn ,Q ωQ n (5.93) Here again, the same as what we had in (5.64), the imaginary part of the ωn summation (which is over bosonic Matsubara frequencies) has a convergence problem that requires us to follow an equivalent of the prescription (5.65). That is achieved by the type of trick used in (5.70) as follows I lim η → 0+ dz κ 2 2 4 ( Q + Q0 ) + z eηz 2) (e βz − 1)(z2 − ωQ " → lim 2πi η → 0+ 1 − β → 1 β ∑ ωn − κ 2 2 4 ( Q + Q0 ) − ω Q 2ωQ (e− βωQ − 1)eηωQ # + Q20 ) + iωn iηωn e =0 2 + ω2 ωQ n κ 2 4 (Q ωn ∑ κ 2 2 4 ( Q + Q0 ) + ω Q 2ωQ (e βωQ − 1)e−ηωQ =0 κ 2 4 (Q + Q20 ) + iωn Q2 + Q20 q = coth 2 + ω2 ωQ n 2Q Q2 + 2Q20 βωQ 2 1 − . 2 (5.94) Comparing (5.94) with the integrand of (5.66), one should realize that they are formally the same apart from the extra factor of coth( βωQ /2) that is present in (5.94). The factor 83 coth( βωQ /2) is a consequence of working at finite temperature and it has crucial effects. Recall that ωQ is the gapless Goldstone mode of the system that goes linearly with Q for long wavelength. On the other hand, coth( x ) is singular at x = 0 and it diverges as 1/x. Consequently, due to the presence of the factor coth( βωQ /2), (5.94) diverges as 1/Q2 as Q → 0. This means that in our two-dimensional system, the integration element of the Q-integral is 2πQdQ and the integral in (5.93) is divergent. In other words, in this limit, long wavelength fluctuations tend to destroy the ordered superconducting state. This phenomenon is a special case of a more general class of phenomena that are described by the Mermin-Wagner theorem. In 1966 in their famous paper, David Mermin and Herbert Wagner showed that in one- or two-dimensional isotropic Heisenberg model, the continuous rotational symmetry can not be spontaneously broken at any non-zero temperature [113]. Later, Sidney Coleman generalized it to any two dimensional field theory with continuous symmetry [40] that also includes our two-dimensional BEC of superconducting pairs where the phase of the order parameter has the continuous symmetry. Speaking of the phase of the order parameter, next we are going to show that although in this limit the order cannot sustain in the Mermin-Wagner theorem sense, the phase of the superconducting order parameter still can have long correlation length. Suppose that the condensate is formed and the symmetry in its phase is spontaneously broken by picking the angle zero, that is to say, ∆0 = |∆0 | (see Figure 5.2). Then fluctuations can be ¯ is seen as small Gaussian noise around the condensate point. In this case, assuming that ∆ conjugate of ∆, we can write ∆ = A + i∆0 P, ¯ = A − i∆0 P, ∆ (5.95) where, as it’s shown in Figure 5.2, A and P are the amplitude and the phase of the fluctuations, respectively. Before starting to see how the spatial correlation of the phase P behaves, note that writing the fluctuation action (5.90) in terms of A and P reveals that it is the excitation modes of P that are gapless and are soft modes of the system while A’s excitations are actually gapped. In fact, simple algebra shows that we can write 84 Figure 5.2. The condensate ∆0 spontaneously chooses its phase to be zero, ∠∆0 = 0. Then P and A are the fluctuations in the phase and the amplitude of the order parameter, ∆, respectively. κ κ ∑ 2∆¯ n ∆n [ 4 (Q2 + Q20 ) − iωn ] + [∆n ∆−n + ∆¯ n ∆¯ −n ] 4 Q20 = n κ κ ∑ 2An An [ 4 (Q2 + 2Q20 ) − iωn ] + 2∆20 Pn Pn [ 4 Q2 − iωn ], (5.96) n where we also used the fact that A and P are real fields and therefore, their real parts in Fourier space are even under inversion n → −n. (5.96) shows how phase fluctuations are gapless while those fluctuations that are trying to change the magnitude of the order parameter are gapped. Now consider the fluctuating phase of the order parameter at two distant points in the sample separated by r = r1 − r2 . We define the spatial correlation between them as 1 i( P(r1 ,τ )− P(r2 ,τ )) he + e−i( P(r1 ,τ )− P(r2 ,τ )) i 2 h( P(r1 , τ ) − P(r2 , τ ))2 i = exp − , 2 hcos( P(r1 , τ ) − P(r2 , τ ))i = (5.97) where for the last step, we used the fact that P is a Gaussian random variable. Now that the task is reduced to calculating h( P(r1 , τ ) − P(r2 , τ ))2 i, next we use (5.95) to write P in ¯ and then going to Fourier space and integrating over the center of mass, terms of ∆ and ∆, a careful calculation gives us h( P(r1 , τ ) − P(r2 , τ ))2 i = 1 2∆20 ∑(1 − cos Q · r)h2∆¯ n ∆n − ∆n ∆−n − ∆¯ n ∆¯ −n i. n (5.98) 85 ¯ n ∆n − ∆n ∆−n − To compute (5.98), first we use our fluctuation action (5.90) to replace h2∆ ¯ −n i in the same way we did in (5.63). Then the ωn summation can be done following ¯ n∆ ∆ the same method introduced in (5.94) to find √ κ 2 2 32Λ ER Λ 2 4 ( Q + 2Q0 ) + iωn h( P(r1 , τ ) − P(r2 , τ )) i = ( 1 − cos Q · r ) (5.99) ∑ 2 + ω2 ∆20 p20 Vβ n ωQ n q  √ Q2 + 2Q20 βω 32Λ ER Λ Q · r Q = coth − 1 ∑ sin2 2  Q 2 ∆20 p20 V Q √ q Z βωQ 8Λ ER Λ p0 2 2 Q + 2Q0 coth −Q , dQ(1 − J0 (rQ)) = 2 π∆20 p20 0 where in the last step, the integral over angle is done that generates the Bessel function J0 . Thanks to the presence of the factor (1 − cos Q · r), the last integral in (5.99) doesn’t have a divergence problem at Q = 0. In fact, one can observe that as Q → 0, its integrand goes like 2r2 Q/βκ. However, for very large Q, the integrand goes as Q20 /Q, which means it can not be extended to infinity and requires us to pick a cut-off momentum. Recalling that we derived the fluctuation action in the limit of Q p0 , p0 is a natural candidate for cut-off momentum. To estimate the final integral in (5.99), first one should realize that beside the p √ small momentum scale Q0 , there is another momentum scale Q1 = 1/ βκ = p0 T/ER that determines the behavior of the occupation factor coth( βωQ /2). In the close vicinity of the transition point when ( Tc − T )/Tc 1, we have Q0 Q1 , and as T approaches to zero when T/Tc 1, Q0 grows and Q1 shrinks until we reach the opposite limit Q1 Q0 . However, since we are working in the vicinity of the transition point, we shall use the former limit Q0 Q1 . In this limit, Q1 itself can be seen as the momentum scale at which coth( βωQ /2) changes behavior; for Q smaller than Q1 , it is in its diverging region and behaves like 2/βωQ , and for Q larger than Q1 , it can be approximated by the flat constant value 1. Therefore, we break the integral into two pieces, for the two distinct behavior of coth( βωQ /2). Then to approximate the resultant integral finally, we are going to assume very large distance limit, rQ1 1, to write q Z p0 βωQ 2 2 dQ(1 − J0 (rQ)) Q + 2Q0 coth −Q 2 0 Z Q1 Z p0 1 − J0 (rQ) 1 − J0 (rQ) 2 2 dQ ∼ 8Q1 + Q0 dQ Q Q 0 Q1 Z Q1 1 − J0 (rQ) ∼ 8Q21 dQ ∼ 8Q21 ln(rQ1 ), Q 0 (5.100) 86 where we drop the second integral in the second line comparing to the first one because it is proportional to Q20 ; and in the last line, we used the limit rQ1 1 to conclude that the integral is dominated by its tail. Using (5.100) in (5.99), we obtain s 32ΛT Λ T 2 2 h( P(r1 , τ ) − P(r2 , τ ))2 i ∼ ln r p 0 . ER ER π∆20 (5.101) Finally plugging (5.101) in (5.97), we find that at large distance, the phase spatial correlation goes as hcos( P(r1 , τ ) − P(r2 , τ ))i = ER T p20 r2 η , (5.102) where the power η is given by 16ΛT η= π∆20 s 48Q21 Λ = ER πQ20 s Λ , ER (5.103) and we used (5.91) to express ∆0 in terms of Q0 . At the transition point, when ∆0 approaches zero, the power η diverges, implying that the correlation is exponentially dying √ out in a length scale determined by Q1−1 = p0−1 ER /Tc . However, as the temperature goes below Tc , the power η decreases and can become of the order of one while the assumption Q0 Q1 is still valid. In fact, (5.103) suggests that when Q21 /Q20 becomes of the order of √ ER /Λ 1, the power η is already of the order of one and hence, the spatial correlation decays like a power law. This transition from an exponentially decaying correlation to a power law decay is known as Kosterlitz-Thouless transition. In 1970 and 1971, Vadim Berezinskii in two seminal papers studied the behavior of the long-range fluctuation correlations in lowdimensional systems at the finite low-temperature limit, where the spontaneous breaking of continuous symmetry had been proved impossible. He considered a classical spin system in the first paper [17] and studied a couple of two-dimensional quantum systems in the second one; an XY model, a 2D Bose liquid, and a 2D isotropic Heisenberg ferromagnet [18]. Soon after him in 1973, John Kosterlitz and David Thouless in their famous paper proposed their new notion of topological order in two-dimensional systems in which no long-range order of the conventional type exists [100], the work that many years later, in 2016, won them the prestigious Nobel prize in Physics. They showed that the new form of order has to do with the interaction between the vortex-like defects that form in the system; vortices that are made out of the two-dimensional arrow of the order 87 parameter and look like whirlpools puncturing the order parameter field, hence the name topological. The physical picture is that vortices of the opposite direction attract each other and their dissociation corresponds to going from power-law correlation to the exponential one. Kosterlitz and Thouless themselves applied their ideas to an XY model, solid-liquid transition, and a neutral superfluid, adding that such transition cannot occur in a superconductor or a Heisenberg ferromagnet [100]. However, later it was shown that in dirty superconducting thin films, when the penetration depth is large enough, the system becomes more like superfluid and goes through KT-like transition [13]. KT-like transitions are also shown in other two-dimensional superconducting systems such as in an array of Josephson junctions [146] or even at zero temperature in a two-dimensional disordered system where the external magnetic field plays the role of temperature [63]. In our case of BEC limit, when the transition is essentially happening between an insulator and a dilute BEC of molecular-like Cooper paris, it is no surprise that we are finding a KT-like transition. To get an estimate of the KT transition temperature, TKT , one can assume that as the temperature goes down from Tc to zero, the square of order parameter’s magnitude, ∆20 , increases linearly from zero to its maximum value, ∆20,max , at zero temperature. This is justified by recalling that ∆20 = −α0 /γ and realizing that around T = Tc , α0 increases linearly from zero. Therefore, we can write ∆20 = Tc − T 2 ∆0,max . Tc (5.104) Then using (5.104) in (5.103) and following the well-known critical exponent result η = 1/8 (for example see Section 3.3 of Ref. [120]), we find TKT = with 128ΛTc c= π∆20,max s Tc , 1+c Λ 32T̃c Λ̃3/2 g 32T̃c g = ≈ , 2 ˜ ˜2 ER κ κ π ∆0,max π∆ 0,max (5.105) (5.106) where we normalized all the energy to Λc0 as we have done in plotting Figure 4.2 and Figure 5.1. For the last step, we considered the fact that we are working close to Λ = Λc0 → Λ̃ ≈ 1. Since close to the critical point Λ = Λc0 , ∆0,max goes to zero faster than Tc , the constant c is generally a large number in the region that we are working. Although one only needs to slightly stay away from the critical point in order to get the constant c of order of one for reasonably small g/κ. 88 To conclude this chapter, we would like to provide a numerical estimate of the transition temperature base on what we found. Assuming a moderate attraction, g/κ = 0.1, we get Λc0 = ER g2 /(16κ 2 ) = 6.25 × 10−3 ER . In order to stay in the region we dis- cussed above, we take Λ = 0.99Λc0 . Then the numerical solution of Eq. (5.72) gives us Tc ≈ 0.22Λc0 ≈ 1.37 × 10−3 ER . That means with a large ER of order of 100 meV, the mean field critical temperature would be about 1.6 Kelvin. Next, to find an estimate for the true transition temperature, which is BKT transition temperature, we shall use our result given by (5.105) and (5.106). To find the constant c, (5.106) requires us to find ∆0,max as well. For that purpose we use the numerical solution of Eq. (4.3) and get ∆0,max ≈ 0.16Λc0 . Then using the same numerical values for g/κ and Λ, from (5.106) we find c ≈ 8.2, which means TKT would be about one order of magnitude smaller than Tc . Finally we would like to compare our result with what He and Huang found in Ref. [83]. The model they use is different as they assume a fixed density and find the transition temperature for increasing value of SOC, whereas in our model, for a positive Λ, density shrinks as we increase the temperature from zero and eventually goes to zero at the transition temperature. In the limit of strong SOC, what Ref. [83] finds for transition temperature in our notation reads as TKT = πnER /8p20 . Using the above values for g/κ, Λ and ∆0,max to calculate density via (4.14), we find n ≈ 0.42 × 10−4 p20 . Based on their formula, this means TKT ≈ 0.16 × 10−4 ER , which is about one order of magnitude smaller than our result. To summarize, we found that in the two extreme limits of our model, fluctuations behave very differently. In the deep BCS limit, in the presence of the ring-like Fermi sea, we found that the fluctuations are damped due to their coupling with the regular conducting electrons at the Fermi level. In the opposite limit, however, at the onsetting point of BEC of Cooper pairs, we found that fluctuations are not damped and the system goes through a KT transition similar to what exists in two-dimensional superfluids for which we provided a numerical estimate. CHAPTER 6 CONCLUSION Starting from a two-dimensional electron gas (2DEG) with a generic Rashba SOC, we set up our single particle playground. Then assuming low-density limit, we simplified the playground further by only keeping the lower energy band, which looks like a moat and hence its name (see Figure 2.2). We were already aware that the peculiar geometry of the moat band makes it possible for the system to go through an insulator-superfluid transition at zero temperature and at some critical attraction [76]. Here the 1D-like diverging density of states (DOS) of the system at its lowest energy plays a crucial role. The enhanced density of states amplifies the effects of any interaction, including the attractive pairing mechanism. However, the triggering idea was that so it (that is the diverging DOS) does the same to the Coulomb repulsion, especially if it’s not well-screened. Therefore, keeping the attractive channel to be a simple contact-like attraction in the Cooper channel, we included the long-range Coulomb repulsion as well. Noting that the diluteness also means a lack of polarization screening, we also assumed an external screening mechanism to form our full Hamiltonian. The somewhat technical point is that since we only consider the lower moat-shape band and it has a spin texture, our projected Hamiltonian acquires phase factors due to the dependence of the electron’s spin orientation on its momentum. In our preliminary analysis, we showed that owing to the diluteness of the system, one can extract many essential features of the system just by studying the problem at the two-body level. Studying the two-body problem without Coulomb, we found the same critical attraction that Chamon et al. had found [76]. Moreover, we found that the inclusion of the Coulomb term in the two-body problem can bring about interesting features to the system. In particular, while even in the presence of Coulomb there is always a critical attraction at which a bound state emerges in the system, we found that the bound state 90 has a peculiar feature in the limit of very week screening. That is it appears with already having a finite size, a feature that is driven by the competition between local attractive g and long-ranged repulsive Coulomb interaction (see Figure 3.2). Aiming to study the many-body aspects of the system, next we used a BCS variational wavefunction. Using BCS wavefunction we were able to extend our problem to the case where there is a shallow Fermi ring in the noninteracting system (that is when Λ is negative in our notation). In the case of neutral fermions, that is in the absence of the Coulomb repulsion we showed that a universal function describes the zero temperature value of the order parameter, ∆, as a function of the location of the chemical potential Λ (see Figure 4.2). More importantly, using this approach, we found that there are two qualitatively different regimes of superconductivity in our model: a regime that can be thought of as a dilute BEC of well-defined Cooper pairs and a regime that resembles standard BCS situation in which the role of Pauli exclusion and Fermi statistics is essential. In the neutral fermion case, a crossover between these two regimes can be observed by changing Λ or the chemical potential. The rather simple zero-temperature phase diagram of the neutral fermion case can be found in Figure 4.1. Studying the effects of Coulomb repulsion within the BCS wavefunction framework, we realized that the electrostatic repulsion manifests itself in two major ways. First it modifies the Cooper channel, and in particular, when it’s not well-screened, it brings internal momentum dependence to the order parameter ∆ p . Second, it creates a not negligible Hartree-Fock term that makes the inclusion of an extra auxiliary field Φ p necessary. Turns out that in this scheme, energy optimization of the system leads to a coupled set of self-consistent equations that connect ∆ p and Φ p (Eq. (4.7)). In the dilute BEC limit, we show that one can construct an approximate solution for this set of nonlinear coupled integral equations using the bound state solution of the Schrödinger equation for the two-body problem (see Section 4.2.2). On the BCS side on another hand, a numerical investigation was required to explore how the transition occurs according to the self-consistent equations. The inclusion of the Coulomb repulsion also makes the phase diagram more complicated. As it can be seen in the Figure 4.3, there is a region of BEC superconductor on the Λ negative side. Moreover, in the presence of the externally screened Coulomb, the gate distance provides another handle to control the 91 electrostatic effects, which makes it possible to move the system to different parts of its phase diagram. The main results of this part of our work were published in a paper in Physical Review B [2]. Finally, we used the path-integral approach to study our model. This approach lets us investigate two other aspects of our model. First, it allows for finite temperature calculation in a very integrated way, and second, it provides a framework to study the effects of fluctuations. After spending some work building the effective action of our model, first, we show how some of our previous zero-temperature results can be derived from the path integral formulation. Then as the first new result, we present a fluctuation analysis at zero temperature for the simple case of neutral fermions. There we conclude that at zero temperature, the ordered phase of our two-dimensional model is robust against the fluctuations. Assuming only the neutral fermion case (which is equivalent to the case of stronglyscreened Coulomb repulsion upon a proper renormalization of the short-range attraction), next we study our model at finite temperature. In the first step at the finite temperature analysis, neglecting fluctuations and focusing on the mean field theory, we derive the mean-field critical temperature of our model, Tc , for different positions of the chemical potential, Λ. Turns out that the mean-field critical temperature, Tc , can be found using a universal function that is plotted in Figure 5.1, which is very similar to what we had for the order parameter at zero temperature, though not exactly the same. Next, we move to study the crucial question of the effects of fluctuations at finitetemperature on our two-dimensional system, a question that ultimately is about the famous KosterlitzThouless (KT) transition. We discovered that the finite-temperature fluctuations of the model close to the transition point show qualitatively different behaviors at the two extreme limits of our model. In the deep BCS limit, it appears that the coupling of the excitations of the superconducting field to the free electrons at the Fermi level damps the fluctuations, similar to what happens to the fluctuations of the superconducting field in the standard theory of superconductivity in regular metals. As the last piece of the present work, in the extreme BEC limit, we find a qualitatively different behavior of fluctuations that resembles that of a two-dimensional superfluid. 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