Dual formulations of non-abelian spin models: local representation and low-temperature asymptotics

TL;DR

This paper develops a local dual formulation for non-abelian SU(N) and U(N) lattice spin models, enabling analysis of low-temperature asymptotics and correlation functions through a Gaussian approximation and semiclassical limit.

Contribution

It introduces an exact local dual representation for non-abelian lattice spin models, extending duality concepts and analyzing low-temperature behavior in detail.

Findings

Dual variables are discrete and live on (D-2)-cells of the dual lattice.

At low temperatures, the dual Boltzmann factor converges to a Gaussian distribution.

The low-temperature limit is described by an ISO(2)-like approximation of SU(2) matrix elements.

Abstract

Non-abelian lattice spin models with symmetry group SU(N) or U(N) can be formulated in terms of link variables which are subject to the Bianchi constraints. Using this representation we derive exact and local dual formulation for the partition function of such models on a cubic lattice in arbitrary dimension D. Locality means that the dual action is given by a sum over some subset of hypercubes of the dual lattice and the interaction between dual variables ranges over one given hypercube. Dual variables are in general discrete-valued and live on (D-2)-cell of the dual lattice, in close analogy with the XY model. We use our construction to study in details the dual of SU(2) principal chiral model in two dimensions. We give dual expressions also for two-point correlation function in arbitrary representation and for the free energy of defects. Leading terms of the asymptotic expansion of the dual Boltzmann factor are computed and it is proven that at low temperatures it converges to a certain Gaussian distribution uniformly in all fluctuations of dual variables. This result enables us to define the semiclassical limit of the dual formulation and to determine an analog of the vortex--spin-wave representation for the partition function. Such representation is used to extract leading perturbative contribution to the correlation function which shows power-like decay at weak coupling. We also present some analytical evidences that the low-temperature limit of the dual formulation is completely described by ISO(2)-like approximation of SU(2) matrix elements.