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Journal of Applied Physics

‘‘Θ physics’’ and quantum spin chains (abstract)

F. D. M. Haldane

Published Apr 15, 1985• 170 citations
Publisher

Abstract

The one-dimensional quantum Heisenberg antiferromagnet with exchange alternation is related to the nonlinear 0(3) sigma model with a topological ‘‘θ term’’ in the Langrangian. The models with parity symmetry (θ=0 or π) are respectively related to the integer and half-integer spin models without exchange alternation. The nonlinear quantum physics of Heisenberg antiferromagnets is placed in a wider context of topologically nontrivial field theories and the spin-Peierls effect in half-integer spin systems (and its absence in integer spin systems) explained.

DOI
10.1063/1.335096

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The one-dimensional quantum Heisenberg antiferromagnet with exchange alternation is related to the nonlinear 0(3) sigma model with a topological ‘‘θ term’’ in the Langrangian. The models with parity symmetry (θ=0 or π) are respectively related to the integer and half-integer spin models without exchange alternation. The nonlinear quantum physics of Heisenberg antiferromagnets is placed in a wider context of topologically nontrivial field theories and the spin-Peierls effect in half-integer spin systems (and its absence in integer spin systems) explained.
The structure of edge states in general finite antiferromagnetic quantum spin chains with arbitrary spin value S is discussed within the framework of the nonlinear-sigma model (NL\ensuremath{\sigma}M) plus a topological \ensuremath{\theta} term. Based on a large-N theory of SU(N) quantum antiferromagnets and strong-coupling expansion, we argue that edge states with fractionalized spin quantum number S' exist in all spin chains with S\ensuremath{\ge}1, with S'=S/2 for integer spin chains and S'=(S-1/2)/2 for half-integer spin chains.
We prove a theorem on the O(3) nonlinear \ensuremath{\sigma} model with the topological \ensuremath{\theta} term which states that the grround-state energy at \ensuremath{\theta}=\ensuremath{\pi} is always higher than the ground-state energy at \ensuremath{\theta}=0, for the same value of the coupling constant g. Provided that the nonlinear \ensuremath{\sigma} model gives the correct description for the Heisenberg spin chains in the large-s limit, this theorem makes a definite prediction relating the ground-state energies of the half-integer- and the integer-spin chains. The ground-state energies obtained from the exact Bethe Ansatz solution for the spin-(1/2 chain and the numerical diagonalizaton on the spin-1,- (3/2, and -2 chains support this prediction.
The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.
Microscopic models of quantum antiferromagnets are investigated on the basis of a mapping onto effective low-energy Hamiltonians. Lattice effects are carefully taken into account and their role is discussed. We show that the presence of an external staggered magnetic field modifies in a nontrivial way the usual mapping onto the nonlinear sigma model, leading to the appearance of terms neglected in previous works. Our analysis is compared with Lanczos diagonalizations of S=1S=1S=1 Heisenberg chains in a staggered field, confirming the validity of the single-mode approximation for the evaluation of the dynamical structure factor. The results are relevant for the interpretation of experiments in quasi-one-dimensional compounds. Microscopic realizations of SU(4) spin chains are also discussed in the framework of spin-orbital lattice systems. The low-energy physics is shown to be described by sigma models with topological angle $\ensuremath{\theta}$ in one dimension. This mapping strongly suggests that the one-dimensional CP3{\mathrm{CP}}^{3}CP3 model (with $\ensuremath{\theta}=\ensuremath{\pi})$ undergoes a second-order phase transition as a function of the coupling.
The relation between quantum spin chains and conformal field theories is reexamined. Using a generalized Hubbard model representation it is argued that the critical theory for generic half-odd-integer spin antiferromagnets is the Wess-Zumino-Witten model (WZW model) with topological coupling, k=1, whereas generic integer spin antiferromagnets have an energy gap. The higher-k WZW models (which describe integrable higher spin models) are multicritical points in the space of all spin Hamiltonians. The k=1 WZW model represents a stable fixed point for many theories including WZW models of arbitrary odd k with relevant operators added, generalized Hubbard or Thirring models with an odd number of colors and the O(3) \ensuremath{\sigma} model at topological angle \ensuremath{\theta}=\ensuremath{\pi}.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.