Operator identity and applications to models of interacting electrons

By correct reduction of some quartic forms in fermion field operators, I eliminate all but a constant and some quadratic terms. I can use this to transform the Wolff model, of a magnetic impurity in a nonmagnetic metal, into a solvable quadratic form in fermions. Applying the same method (with less justification) to Hubbard's model in three dimensions, I obtain an oversimplified but nevertheless suggestive, and diagonal, Hamiltonian.

Operator Identity and Applications to Models of Interacting Electrons Daniel C. Mattis Belfer Graduate School of Science, Yeshiva University, New York, New York 10033 (Received 21 July 1975) By correct reduction of some quartic forms in fermion field operators, I eliminate all but a constant and some quadratic terms. I can use this to transform the Wolff model, of a magnetic impurity in a nonmagnetic metal, into a solvable quadratic form in fermi­ons. Applying the same method (with less justification) to Hubbard's model in three di­mensions, I obtain an oversimplified but nevertheless suggestive, and diagonal, Hamil­tonian. In this paper I present elements of a novel so­lution of a well-known model magnetic impurity in a nonmagnetic metal, first proposed by Wolff.l In an earlier study Tomonaga's method2 was used to reduce this many-body problem to a solvable quadratic form in boson operators.3 I have now transformed it further, into an exactly solvable quadratic form in fermion operators, by a meth­od detailed below. Like the old, the new method indicates that there is a singularity at a value of the Coulomb parameter U c of the order of the bandwidth, but unlike the 0ld4 it can provide de­tailed results in the critical region U"" U c replete with symptoms of the Kondo effect: specific-heat anomaly, high magnetic susceptibility, low-fre­quency resonance which can translate into a re­sistance anomaly. Finally, an apprOXimate ex­tension of this method to Hubbard's model5 of an interacting electron gas yields as the solution of that problem a Simple, diagonal, yet physically plausible Hamiltonian. I recapitulate parts of my earlier study. 3 The interaction between electrons of opposite spin (J at the impurity site R j =0 is given by ~ = U[n,(O) - t)[n, (0) - tl =}U{[nt(O) +nl(O) -1]2 - [nt(O) - nl(O)]2}. (1) The dual representations of the local occupation­number operator are (2a) which is quadratic in the fermion field operators Ck, and na(0)-i=N- 1 L; (Pqa +P qa t), a >0 Pqa =P-aa t =L;Cka t Ck+ q•a, k (2b) which is linear in the bosons. Momentary reflec­tion suggests that Je2 , expressed in bosons, will become separable after a unitary transformation to new operators denoted Par=2-1/2(Pqt+TPal), where T = ± 1. The "kinetic energy" operator Jeo of a half-filled band is invariant under such trans­formation. Originally there are two equivalent forms: Jeo=L; €kCkatcka' Ek=vF(k-k F), k.a in the original fermions, or Jeo=27TV FN- 1 L; p.a t paa , q > O,a (3a) (3b) in the original bosons. After transformation to the new operators, PaT> this last equation be- 483 VOLUME 36, NUMBER 9 PHYSICAL REVIEW LETTERS 1 MARCH 1976 comes (4a) which, in turn, possesses an equivalent form in a new set of fermion "quasiparticles" ckn viz., Jeo- 6 Ek CkT t CRT • (4b) k, r We have now achieved a separation of variables, Jeo +Je2 - Je+ +Je_, Je r =6 Ek Ckr t Ckr +~ TU[nr(O) - ~F, k T=± 1, (5 ) and nr(O) - ~ given in Eqs. (2) with T replacing a henceforth. The original work was based on rep­resentations (4a) above; the present goal is to make use of (4b). The new approach must over­come an obstacle that, according to the usual trivial identities, [nr(O)-~F=i. This implies that the interaction Hamiltonian is a trivial con­stant, contrary to the results obtained by use of Tomonaga operators. The following section is devoted to a resolution of this paradox. It is known that for boson and fermion repre-sentations to be unitarily equivalent, certain stability requirements must be imposed.2 One of these requirements concerns the normal or­dering of electron operators, with the filled Fer­mi sea as the vacuum, required in order to prove such important commutation relations as [pq, TO P- q', r'] = (Nq/2rr)6 qJ) TT ,. In the present context, I have found that normal ordering of the boson operators is also required. Indicated by the conventional colon pairs, it is defined as fol­lows: :PexPst:=Pstpex, (6) Normal-ordering eliminates from the expansion of [nr(O) - ~F only those undesirable terms such as PqrPqr t which can contribute large numbers to the vacuum energy, while, at the same time, preserving all the operator equations of motion upon which the earlier solution3 was based. The effects of this procedure can only be gauged after the reintroduction of quasiparticle operators ck into (5); e.g., :[n(O)_~p:=N-2 6 (p/pq,t +p/pq,)+H.c., q,q' > 0 which, with the aid of (2b), becomes :[n(O)-~F:=N-2 6 6 (c/c2tc1tco+H,c,)+N-2 6 6 (c/c2c1 t co +H.c.). (7) 3>21>0 3>21>0 For brevity, k i is simply denoted by i, Eki> Eki by i> j, and the subscript T is omitted. One now needs the integrated density-of-states function S(E), defined as S(Ek )=N- 1 6, €k,<E k 1 = J Ek dE''J((E'), ( min (8) with 'J((E) the density-of-states function, normalized to 1 per unit cell. After some additional algebra­ic drudgery, one obtains : In(O) - ~]2: =N- 16 [S(E3) - S(E F)] [c/ Co + H,c,] +N- 1S(E I') 6 (c3 t Co + H.c.) 3,0 3,0 The last term, quartic in the c's, vanishes by antisymmetry. Because S(E F) = ~ in a half-filled band, the second and third terms cancel, with the fourth term precisely t. It is the first term, quadratic in the field operators, which is the prinCipal consequence of the ordering procedure, Omitting the can­celed terms and generalizing to an arbitrary point R n, one obtains the new result, :[n(Rn)-~J2: =N- 1 6lS(E,,)-S(EF)j{cktck,exp[i(k -k')Rn]+H.c.}+i, (lOa) k,k I as the alternative formulation of : In(Rn) - t]2: =N- 2 6 {exp[i (q +q' )Rn ]Pq t pq , t + exp[i(q - q' )Rn ]Pq t po'} + H.c. (lOb) q,q'>o Summing these over N equally spaced pOints Rn on a line and equating, one obtains the following rela- 484 VOLUME 36, NUMBER 9 PHYSICAL REVIEW LETTERS 1 MARCH 1976 tion: L; [S(E k) -S(EF)Jc k tC k "'N-1L;pq t pq • (11) k q >0 For 10 k "'vF(k -kF ) and a constant density of states ~l= (27TVF)-\ this relation may be seen also to pro­vide a new proof of the equivalence of the two forms of kinetic energy (3a) with (3b), and (4a) with (4b). With (lOa) substituted back into the JeT> Eq. (5), this Hamiltonian is now quadratic in the C k field operators and therefore exactly solvable, Whereas a study of J( _ brings out an interesting singularity, Je+, which can be studied in an analogous manner, demonstrates few noteworthy features for U >0 and can be omitted in the present brief summary. Now, in a magnetic field Je_ must be augmented by the Zeeman energy - h[n t (0) - n I (O)J- - h21/2[n _ (0) -!J and becomes Je_ == L;EkC k- t Ck_ -! UN- l2] [S(E k ) - S(EF)][C k-t C k' _ + H.c.] - h21/2[n _(0) - !J. (12) k ~k' Suppose one studies the Wannier operator at the R j == ° site, aD = N-1/2L;ck_' by means of its equation of motion: (13) One finds that, for a constant density of states~, aD becomes decoupled from the remaining fermion operators at a value of U to be denoted U c, and given by 1 -!U c~ == 0. For a calculation of the tem­perature- dependent susceptibility at U* U c it is more convenient to adopt a unit bandwidth and a semi­circular density of states, 10 F == 0. After some analysis one obtains (14) with f({3E) the Fermi function at temperature T =={3-l, E = -i(z +Z-l), L(U)=2U /U c- (U /U d, and the indicated contour being the unit circle. At T '" ° this yields (15) The zero-temperature susceptibility blows up at U c. The high-temperature susceptibility approaches the free value 1/ 4T for all U. For the study of thermodynamic functions such as entropy the most compact expressions involve the phase-shift function cp(E, U) = tan- 1[(1 - 4E2)1I2(1 -U /U d2( - 2Et 1 (1 +L t 1 J , (16) or rather, the incremental phase-shifts t..cP = cp(E, U) - cp(E, 0). For the incremental entropy t..S(T, U) = S(T, U) - S(T, 0) one obtains t..S (T, U) '" - (7T T r 1 J dE [8f( {3E )/8E JE t..CP(E, U) • (17) In zero magnetic field at T '" 0, a partial integration reduces this to GO \OforU*Uc A.S(O, U) = 2t..cp(0, U)7T- 1 10 dx f(x) "') ln2 at U ",U c (18) In the critical region U ~ U c the incremental entropy vanishes at T '" ° but rises rapidly to a magnitude approaching ln2 at a temperature T m of the order of (1 - U /U d2. The specific heat, given by t..c (T, U) '" - (7T Tt 1 J dE E2[8 f( {3E )/8 E] (8t..cp /8E), (19) also has a sharp, linear, initial rise when U is in the critical region, and displays a maximum at a temperature 0 (T m)' This behavior is a consequence of the rapid variation of t..cP in a region ± T m of the Fermi energy. It is therefore reasonable to suppose that the electrical resistance will also drop for T > T m, and that T m is approximately the Kondo temperature. In the absence of the very complicat­ed, detailed calculations required for the electrical resistance, there is not much more to add on this score. But it is amusing to compare, in the limit U -U c when both A.c(T)/T and X(T) become infinite at low temperature, their ratio, which remains finite: lim(T - 0, U - U dt..c /XT '" 27T2 /3, in accord with Wilson's recent result6 in the Kondo problem. The detailed comparison with other models exhibiting 485 VOLUME 36, NUMBER 9 PHYSICAL REVIEW LETTERS 1 MARCH 1976 the Kondo effect, and detailed derivation of the results quoted above, will be published elsewhere. The three-dimensional Hubbard models incorporates interactions U[nt(R;)-i][nj(R;)-i] at every site R; of a three-dimensional lattice. Because the Tomonaga operators defined on the ith site do not commute with those on a jth site, the transformation to the ± operators is no longer exact. Ignoring the correction terms, we avail ourselves of (lOa) and, after summing over sites, obtain (20) Note that the parameter iu of Eqs. (12) et seq. is now replaced by U. As U is increased, the suscep­tibility rises until it diverges at U 0' given by (21) which is the usual ball-park figure for a phase transition. The bandwidth of the (+) quasiparticles has approximately doubled at U o' Unlike the reduction of the Wolff model, the result (20) is not exact. Nevertheless, it contains all the qualitative features one expects, except for the finite lifetime of quasiparticles away from the Fer­mi energy. Many such aspects of the problem appear to be of interest, and further details will be pub­lished in due course. Ip. A. Wolff, Phys. Rev. 124, 1030 (1961), and solved by him in the Hartree-Fock approximation. 2S. Tomonaga, Prog. Theor. Phys. Q, 544 (1950). A review can be found in E. Lieb and D. Mattis, Mathematical Physics in One Dimension (Academic, New York, 1966), Chap. 2. More recent applications and extensions are given in K. Schotte and U. Schotte, Phys. Rev. 182, 479 (1969); D. Mattis, J. Math. Phys. (N. Y.) 15, 609 (1974); A. Luther and V. Emery, Phys. Rev. Lett. 33, 589 (1974), inter alia. 3D. Mattis, Ann. Phys. (N. Y.) 89, 45 (1975). 4At high temperature or with strong interactions, the number of degrees of freedom associated with bosons great­ly surpasses that for fermions so that correct thermodynamic properties near Vc are difficult to extract from a pure Tomonaga model. Criteria for the validity of the Tomonaga operators have been given, e.g., by M. Aizenman and H. Gutfreund, J. Math. Phys. (N. Y.) 15, 643 (1974). 5See the review by C. Herring, in Magnetism, edited by G. Rado and H. Suhl (Academic, New York, 1966), where the model and several alternative solutions are compared. 6K. Wilson, in Nobel Symposia. Proceedings of the 1973 Symposium, edited by B. Lundqvist et ai. (Academic, New York, 1974), p. 68; K. Wilson, Rev. Mod. Phys. 47, 773 (1975).