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Physical Review Letters

Nonlinear σ model at θ=π: Euclidean lattice formulation and solid-on-solid models

Ian K. Affleck

Published May 13, 1991• 53 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.66.2429

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Physical Review Letters•May 17, 2013•42 citations
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.
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.
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.
On the infrared limit of the O(3) nonlinear σσσ-model at θ=πθ= πθ=π
2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.
The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.
We study the effects of a topological Theta term on 2+1-dimensional principal chiral models, which are nonlinear sigma models defined on Lie group manifolds. We find that when Θ=π, the nature of the disordered phase of the principal chiral model is strongly affected by the topological term: it is either a gapless conformal field theory, or it is gapped and twofold degenerate. The result of our Letter can be used to analyze the boundary states of three-dimensional symmetry protected topological phases.