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We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way.","first_page_t":"749","rights_management_t":"(c)American Institute of Physics. The following article appeared in Lieb, E., & Mattis, D., Journal of Mathematical Physics, 3(4), 1962 and may be found at http://dx.doi.org/10.1063/1.1724276","title_t":"Ordering energy levels of interacting spin systems","journal_title_t":"Journal of Mathematical Physics","id":705215,"publication_type_t":"Journal Article","parent_i":0,"type_t":"Text","subject_lcsh_t":"Magnetism, Band theory of; Ferrimagnetism; Antiferromagnetism; Spin waves","thumb_s":"/c6/f3/c6f3ebca36d0cd5401eb3ce7f54097f69351bf1e.jpg","last_page_t":"751","oldid_t":"uspace 3045","metadata_cataloger_t":"CLR; KWR","format_t":"application/pdf","modified_tdt":"2012-06-13T00:00:00Z","school_or_college_t":"College of Science","language_t":"eng","issue_t":"4","file_s":"/3f/99/3f99ecdfc71b3888ad68f0c0a871f280b9e70aea.pdf","format_extent_t":"253,796 bytes","citatation_issn_t":"222488","other_author_t":"Lieb, Elliott","created_tdt":"2012-06-13T00:00:00Z","_version_":1664094540378669056,"ocr_t":"JOURNAL OF MATHEMATICAL PHYSICS VOLUME 3, NUMBER 4 JULY-AUGUST 1962 Ordering Energy Levels of Interacting Spin Systems ELLIOTT LIEB AND DANIEL MATTIS Thomas J. Watson Research Center, International Business Machines Corporation, Yorktown Heights, New York (Received October 6, 1961) The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) 1 is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S ~ O. In ferrimagnetism, E(S + 1) > E(S) for S ~ S, and in addition the ground state belongs to S :$ S, S is defined as follows: Let the maximum spin of the A sublattice be SA and of the B sublattice SB; then S == ISA - SBI. Antiferromagnetism is treated as the special case of S = O. We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field. INTRODUCTION THE general Heisenberg Hamiltonian for interacting spins on a lattice (in any number of dimensions) is (1) This describes theories of ferromagnetism, ferrimagnetism, and antiferromagnetism, depending on the geometry of the lattice, the structure of the symmetric matrix J ii, and the magnitude of the intrinsic spins (which may vary from site to site). In fact, it is conceivable that these factors be such that the spin system displays a mixture of the three magnetic properties. But we shall restrict the discussion to ferrimagnetic arrays, of which a special case is anti ferromagnetism. We consider only those arrays for which an A and a B sublattice can be defined. The definition of these two sublattices is circular, and perhaps not unique, for the only requirement in defining them is that there exist a constant l 2: ° such that for all sites i(A) on one sublattice and i(B) on the other, and J,(A),i(B) 2: g2. (2) In general, there might be several ways to decompose the lattice in such a way that (2) is obeyed, or there may be none. In the latter case, the system is not necessarily ferromagnetic, and only explicit solutions will reveal its properties. But if (2) is obeyed, we shall show that one is definitely dealing with ferrimagnetism or antiferromagnetism. Note that the number of sites in each sub lattice and the magnitude of the intrinsic spin on each site is irrevalent, so that only the topology of the lattice and the structure of J;; counts. Note also that for g = 0, and the A sublattice consisting of the nearest 749 neighbors to the sites on (i.e., intermeshing with) the B sublattice, the requirement (2) gives a tendency for nearest neighbors to align antiparallel and next-nearest neighbors to align parallel, and therefore reduces to the usual definition of ferrimagnetism (when the spins are of unequal magnitude) and of antiferromagnetism (when all spins are equal). The intrinsic spin of an electron is 1/2, but we may be dealing with various species of magnetized atoms or nuclei, so let the intrinsic spin angular momentum on each site be Si' The maximum possible spin S A on the A sublattice is therefore (3a) and on the B sublattice (3b) Defining (3 c) we shall prove that the ground state of H belongs at most to total spin S = S. Moreover, if we denote by E(S) the lowest energy eigenvalue belonging to total spin S, then we shall also prove E(S + 1) > E(S) for all S 2: S, and (4) E(S) > E(s) for S < Sand l = 0. (Antiferromagnetism is when S = 0, and the ground state belongs to total spin zero.) This can be regarded either as a theorem in ferri- or antiferromagnetism, or as a proof that the conditions in Eq. (2) and above eliminate the possibility of ferromagnetism (insofar as it costs energy to raise the total spin value over and above its ground-state value, and that this ground-state value is far from the maximum per- 750 E. LIEB AND D. MATTIS missible value of SA + SB). It also indicates that a large class of apparently different Hamiltonians (1) have really a similar structure, as summarized in Eq. (4), and in the properties of the corresponding eigenfunctions which we shall find below. W. Marshall was the first to showI that the ground state of an antiferromagnet is a singlet; Elsewhere,2 we have commented on and strengthened his proof. In the present work, we succeed in removing the requirement of translational invariance, and also apply the method to identify the excited states. The M-subspace arguments presented here were previously found useful in the classification of the states of an electron system, and have been used to disprove the possibility of ferromagnetism in linear chains of atoms in 8 states. 3 We shall now restrict the discussion to the special case l = 0, until the end of the proof. M SUBSPACES With the help of the total spin operator 8 == L: S, we can construct two operators which commute with each other and with H, namely, 8 2 and 8\" which possess eigenvalues S(S + 1) and M, respectively. It is known from the theory of angular momentum that S 2:: IMI. From the rotational invariance of the Hamiltonian we infer the (2S + I)-fold degeneracy of each energy level belonging to S, one degenerate level for each value of M in the range -S :::; M :::; S. It therefore follows that every energy eigenvalue has a corresponding eigenfunction (representative) in the M = ° subspace of eigenfunctions; that every energy level except those belonging to S = ° has a representative in the M = 1 subspace; similarly for all except S = ° and S = 1 in the M = 2 subspace, and so forth. The theorem, Eq. (4), will be proved if we can show that the lowest energy in an M subspace belongs to S = M, for spin S + 1 also has a representative in that subspace and therefore E(S) < E(S + 1). If the ground state belongs to S = So (we still have to prove that So :5 S), we need only consider the subspaces of IMI 2:: So, for the ground states of the remaining subspaces will always belong to So. The mechanics of the proof are this: The ground state of H in an M subspace is not orthogonal to the ground state of a soluble Hamiltonian in the same subspace, and the latter is known to belong to 1 W. Marshall, Proc. Roy. Soc. (London) A232, 48 (1955). 2 E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16,407 (1961), particularly Appendix B. 3 E. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962). S = M for M 2:: S; therefore, so does the former. N ow let us go into more detail. PROOF In an M subspace, choose the basis set to consist of all distinct eigenfunctions of the s~ compatible with eigenvalue M. We denote each configuration in the set by rJ>., where a is an index which runs over all members of the set. Shortly, we shall specify a convenient choice of phase for each configuration. But first, perform a canonical transformation on H by letting 8~(A) ~ +8~(A) (5) but leaving the spins on the B-sublattice invariant. In the new language, the Hamiltonian can be written as Ho + HI' where the diagonal part is Ho = 2 L: Ji;8~8;, and the nondiagonal part is (6) HI = - {L IJiil S:Si + H.c.}. (7) We recall that l of Eq. (2) is zero: the generalization for g2 > ° comes below. In a given state rJ>., S~ has eigenvalue mi' Choose the phase of rJ>. in the following manner: rJ>. = C(s~)S,+m,(s;)S,+m, ... (S;')SN+mNx, (8) where X is the state in which m, = -S\" and C is a positive normalization constant. With this definition in mind, it is clear that if we define K~e to be (9) then K~. :5 0, or equivalently, K~a = -IK~.I. (10) The ground state in the M subspace is denoted 1/1, belongs to the ground-state energy EM, and can be expanded in our complete set in terms of the amplitudes fe, (11) Since Ho is diagonal, denote its eigenvalues by ea, (12) and therefore the Schrodinger equation reads - L: IK~al'f~ + eat. = EMf.· (13) ~ The varia,tional energy of any trial function exceeds EM, unless it is also a ground-state eigenfunction. ORDERING ENERGY LEVELS OF INTERACTING SPIN SYSTEMS 751 But vI' = I: Ifal cf>a (14) is a trial function with variational energy EM, and therefore - I: IK\"allf,,1 + ea Ifal = EM Ifal. (15) Moreover, ea - EM > 0, for all a (16) (otherwise, some one cf>a would be the ground state, which is in general impossible.) Therefore, taking the absolute value of (ea - EM)fa as given be Eq. (11) and combining with Eq. (15), we obtain I(I: IK\"al f,,)1 = I: IK\"allf\"l· (17) This is a contradiction unless f\" 2:: 0 forall {3. (I 8) In general, we have a slightly stronger result, f\" > 0, for all (3. (19) For, if some fa vaaished, then Eq. (15) would read: I: K\"a If,,1 = 0, and by succeeding applications of the Hamiltonian, one could establish that all the amplitudes vanished, unless the Hamiltonian splits into sets of noninteracting spins in which case only the weaker result (18) holds. Therefore, in general, all amplitudes are positive and nonvanishing, and hence EM in nondegenerate. This last statement follows from the impossibility of constructing states orthogonal to 1/1 without some changes of sign, and consequent violation of the ground-state property (19). Next consider the special Hamiltonian where J;(Ali{Al = J;(Bl;(Bl = 0 and J;(Al;(Bl = J, a positive constant. The eigenvalues are readily calculated. The lowest energy belonging to each spin is given by E(S), for S 2:: S, and the ground state belongs to S = S. E(S) = J{S(S + 1) - SA(SA + 1) - SB(SB + I)} for S 2:: s. (20) By the previous arguments, the ground-state eigenfunctions of this special Hamiltonian in a given M subspace satisfy Eq. (18) or (19) and are therefore not all orthogonal to the corresponding ground state of H. The special Hamiltonian has an S = M ground state in each M subspace, provided M 2:: S. Therefore, so does H and this completes the proof for l = O. When l > 0, we have proved the theorem (4) for H - lS2 and it is therefore true a fortiori for H. However, the lowest ground state no longer necessarily belongs to S, but belongs to S ~ S. MAGNETIC FffiLD A magnetic field in the z direction but of arbitrary and variable amplitude B; modifies Ho but not HI, and therefore (18) or (19) are still valid for the ground state in an M subspace. The absolute ground state of the system is no longer necessarily in the M ~ S subspace nor is S a good quantum number in the presence of such a magnetic field. ACKNOWLEDGMENT It is a pleasure to thank Dr. T. D. Schultz for helpful discussion."}]},"highlighting":{"705215":{"ocr_t":[]}}}