Spin ordering in three-dimensional crystals with strong competing exchange interactions

Abstract

In this paper we discuss magnetic ordering in three-dimensional crystals in which Heisenberg exchange interactions appear to dominate. Particular emphasis is placed on systems with strong competing exchange interactions, called frustrated spin systems. In the absence of such competition, one finds collinear spin ordering. However, strong competing interactions lead to non-collinear structures, often spirals and many variations thereof. The problem of understanding the origin and physical properties of such spin states within the classical Heisenberg model has been intensively addressed by a large community of researchers over the last 2–3 decades. The study of such problems actually began about five decades ago, and led to a large body of literature that has been overlooked in the publications of the last 2–3 decades (with two very recent exceptions). The early work established important fundamental concepts, including an unconventional theoretical approach, the generalized Luttinger–Tisza method and the idea of forced degeneracy. This resulted in the spin state called a ferrimagnetic spiral (FS), which enabled understanding of puzzling neutron diffraction data, plus NMR and ESR measurements, on the spinels XCr₂O₄, X = Mn, Co. Additional new aspects of this magnetic ordering have been uncovered in two very recent experimental works, indicating a partial spin-liquid-like behaviour and production of ferroelectricity by the pure spiral component of the FS. This exciting relevance of the early work to these modern results and the general lack of awareness of the early work has provided a source of motivation for this review, which covers that early body of work, both theoretical and experimental, and which also connects to the recent studies. A thorough discussion of the calculations, and comparison with experiments is presented. It will be seen that fundamental puzzles remain for these systems.

Keywords:

• Magnetism
• frustration
• competing interactions
• spirals
• spin liquids
• phase transitions
• spinels

Acknowledgments

One of us (T.A.K.) thanks S. D. Mahanti for many helpful discussions concerning the physics and the philosophy of the spin-ordering problem; also P. Duxbury, J. Bass, C.-Y. Ruan, S.-W. Cheong, and P. Horsch contributed useful comments. We thank J. Hastings for a helpful remark concerning his paper with L. Corliss, and to Y. Tokura for a similarly helpful remark about reference 33. We are indebted to K. Tomiyasu for extensive communications concerning his experimental work and its interpretation, plus a critical reading of the manuscript. We also thank H. J. Zeiger for reconnecting us to his work on ESR, K. Siratori for communications concerning his ESR studies, T. Tsuda for sending us and discussing reprints of his NMR papers, and C. Broholm for explaining his papers on ZnCr₂O₄. We thank H. E. Stanley for many valuable suggestions for improving the manuscript, and D. C. Mattis and A. B. Harris for their correspondence. The considerable help in technical aspects of producing the manuscript given by Mr. Christian Hicke is gratefully acknowledged.

Notes

†See Appendix A for a proof of this long-known fact.

†Kaplan 8 showed that similar degeneracy occurs for Heisenberg spins on the B-sublattice. Wannier 9 showed that, similarly, there is macroscopic degeneracy in the ground state of the Ising model on the triangular lattice with antiferromagnetic nearest-neighbour interactions. But it is rather widely believed that there is no such degeneracy if the Ising spins are replaced by Heisenberg spins.

†Use MatchQ[expr,Rational[,]] in Mathematica.

†We find it curious that this analytic method was used to yield “minimum-energy modes”, but that it apparently was necessary to resort to “direct numerical minimization of the energy for small systems of size multiples of 12” to obtain the detailed spin state.

†For the materials discussed here, distortion of the crystal structure due to spin–latttice interactions or magnetostriction have been too small to be seen.

†The theoretical discovery of spiral or helical spin states was made independently by Kaplan 10, by Yoshimori 62, and by Villain 63. The only rigorous proof that such states give the minimum energy was reported in 10 for the specific case mentioned. (T.A.K. thanks P. W. Anderson, private communication, circa 1959, for suggesting a possible proof by the LT method.) Villain considered only local stability. Yoshimori's formalism was strongly similar to that of the LT approach; however, the logic of the LT method was neither explicitly invoked nor appreciated (see his footnote on p. 809, where a more general constraint is considered; this is superfluous if one recognizes the LT logic, and raises the possibility that one of an infinity of even more general constraints might give a lower energy, impossible by the LT argument). A quite misleading account of the history of the spiral discoveries is given in 64, where the following is stated, p. 64: “In his 1942 PhD thesis at MIT, J. M. Luttinger pointed out” that “if j(q) has a minimum at” … “q ₀, then among the ground state configurations one must include the spiral configuration '. There is also an absence of reference to the contributions (1959) of Kaplan, Yoshimori and Villain. The clear implication is that the spiral idea appeared in Luttinger's thesis (so the later discoveries are superfluous). If true, this would be important to know. However it is not true. I (T.A.K.) have read through the thesis; there simply is no such statement. Dr. Mattis replied to my inquiry about this that he had relied on oral transmission from Luttinger. Furthermore, we are not aware of its publication elsewhere by Luttinger. Incidentally, the correct date for the submission of the thesis is 1947 (see 15). We add here that errors in 64 concerning the interpretation of the LK theorem (p. 13) are described in footnote ‡, p. 13 (present paper).

†This is stronger than necessary for the theorem to hold. For example, this symmetry need only hold for the Hamiltonian (4), wherever non-magnetic ions may be located.

†This theorem was incorrectly attributed to the Luttinger–Tisza paper in a recent work 51, p. 231. As just discussed, LT discussed a lattice of dipoles, and considered only states that allow for a small number (≈ 10) of spin degrees of freedom, in contrast to this theorem, which is a statement about the minimum energy over the full problem of ≈ 10²³ spin degrees of freedom.

‡In addition to the error cited in the footnote on p. 11, misinterpretations of these results occur in 64. There, on pp. 289–290 it is stated incorrectly that LK considered only nearest-neighbour interactions J_ij . Also it is stated there that the same method works even for non-Bravais lattices; that this might not be true in general, even for Heisenberg spins, was discussed above and is seen explicitly in section 4 of the present paper.

†An alternate method, already mentioned, is described in 46, where it is used to solve rigorously and analytically many such problems.

†This is true for the functions of interest here. See 65.

†Yoshimori 62 and Anderson 40 mention formal aspects of this approach. Anderson incorrectly states 40, p. 124, this ‘is Kaplan's method’ (with reference to LK and KLDM); see footnote, p. 11 for a comment on Yoshimori's usage.

† are orthonormal unit vectors unrelated to the crystalline axes (because of the spin rotation symmetry of H).

‡We loosen the definition (3) to allow k ₀ to change with u as u increases past u ₀.

†It is expected to hold for the rather large spins involved in our present cases, but see 75.

†A technical detail: The having cubic symmetry implies that the eigenvalues will be invariant under a cubic symmetry operation on k. Alternately, if is the 6N× 6N matrix consisting of the matrices , along the diagonal, then the usual definition of cubic-symmetry invariance, the commutation of with any cubic operation, holds.

†This is true if k is isolated.

‡We refer the reader to LKDM for a derivation of the magnetic intensity formulas appropriate to the FS as well as more general structures.

†The equation τ =0.983, in the caption and in the text, p. 563 of HC, left column, is a misprint. It should be τ=0.0983 (Julius Hastings, private communication, 2005), as corrected here.

†We thank K. Tomiyasu, private communication, for this explanation.

†We have been frustrated by not being able to find in the literature a definition of this term, particularly one that distinguishes the term from frustration (without the modifier, geometric). A long time before the term geometric frustration was introduced, a definition of ‘frustration’ appeared in the literature in a paper by Toulose 83. This includes the above meaning of geometric frustration as the special case where there is macroscopic degeneracy. Toulose's definition is identical to ‘competing interactions’, in the sense of classical spins, as precisely defined in 16.

†ZnCr₂O₄ apparently is an example where spin–latttice interactions remove the degeneracy: there is a transition to tetragonal lattice structure at 12.5 K, with decreasing T, the spin state changing from being short-range ordered to a long-range-ordered Néel state 86. A later reference 87 does not mention this transition. Apparently (pure) ZnFe₂O₄ does show only SRO at very low T 88.

†A similar suggestion was made by HC 19.

†Stickler 28, first describes the anomaly at ∼20 K as follows: At 20 K, mode C fades out and mode D appears at 42 GHz, remaining there down to 4.2 K. He then describes it as a sharp rise from 16 to 42 GHz. It is reminiscent of a first-order phase transition.

†Actually the best that can be done is to have k ₀ and -k ₀ domains, i.e. within the Heisenberg model, such domains are degenerate. This would not spoil the argument. An anisotropic term of the Dzyaloshinskii–Moriya form could remove this degeneracy.

†Just before we were to submit a revised version of this paper, we noticed that Yamasaki et al. 33 found, in CCO, an anomaly in the magnetization and electric polarization at about 15 K, about half way between the jump in the uniform resonance frequency that occurs in the experiments of Stickler et al. 28 and Funahashi et al. 31, and attributed to ‘the lock-in’, with reference to Funahashi, Morii et al. 93. However, the results of the latter work have been strongly contested: see 32; also 80. There still remains the possibility that the anomalies seen are related to the jump in resonance frequency, but the interpretation remains unclear.

†A possibility not mentioned is that the molecular field might vanish, invalidating the argument. However, by Nernst's theorem, there will always be some neglected interactions (including e.g. spin–latttice interactions) that will remove the implied degeneracy; then taking such interactions to 0, one is still left with the classical model.

†We add that the latter type of terms also occur for S=1/2, so there really is no need to go to large S to invoke these types of criticisms. Also, these are all operators allowed by the group of uniform rotations of the spin system.