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Physical Review Letters

Ground-state energies of the nonlinearσmodel and the Heisenberg spin chains

Shoucheng Zhang, Heinz J. Schulz, Timothy A. L. Ziman

Published Sep 4, 1989• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.63.1110

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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 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 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.
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.
The Heisenberg spin ladder is studied in the semiclassical limit, via a mapping to the nonlinear \ensuremath{\sigma} model. Different treatments are needed if the interchain coupling K is small, intermediate, or large. For intermediate coupling a single nonlinear \ensuremath{\sigma} model is used for the ladder. It predicts a spin gap for all nonzero values of K if the sum s+s\ifmmode \tilde{}\else \~{}\fi{} of the spins of the two chains is an integer, and no gap otherwise. For small K, a better treatment proceeds by coupling two nonlinear \ensuremath{\sigma} models, one for each chain. For integer s=s\ifmmode \tilde{}\else \~{}\fi{}, the saddle point approximation predicts a sharp drop in the gap as K increases from zero. A Monte Carlo simulation of a spin-1 ladder is presented which supports the analytical results.
We study the strong-coupling expansion of a Euclidean lattice formulation of the two-dimensional O(3) \ensuremath{\sigma}-model at nonzero topological angle \ensuremath{\theta}, mapping it onto a solid-on-solid model with well-known critical behavior. The results are consistent with those predicted by the mapping of the Hamiltonian formulation of the \ensuremath{\sigma} model onto quantum spin chains. The model has a gapless phase at \ensuremath{\theta}=\ensuremath{\pi} and weak coupling in which a condensate of Coleman's ``half-asymptotic'' particles exists.