Quantum Anisotropic Sigma and Lambda Models as Spin Chains

TL;DR

This paper quantizes anisotropic lambda and sigma models as spin chains, solves them via Bethe Ansatz, and explores their spectra, S-matrices, and continuum limits, revealing insights into their effective field theory regimes.

Contribution

It introduces a lattice Hamiltonian formalism for anisotropic deformations of SU(2) models as spin chains and analyzes their spectra and S-matrices, connecting to known sigma models.

Findings

The spin chain relates to the higher spin XXZ Heisenberg chain.

The S-matrix in the gapped regime is explicitly derived.

The lambda model spectrum matches the O(3) sigma model in a certain limit.

Abstract

We consider lambda and anisotropic deformations of the SU(2) principal chiral model and show how they can be quantized in the Hamiltonian formalism on a lattice as a suitable spin chain. The spin chain is related to the higher spin XXZ Heisenberg chain and can be solved by using the Bethe Ansatz. This yields the spectrum and S-matrix of the excitations. In particular, we find the S-matrix in the gapped anti-ferromagnetic regime. In this regime, a continuum limit does not exist and this suggests that the field theories in this regime, precisely ones with a cyclic RG like the Yang-Baxter deformations, may only exist as effective theories. In a certain limit, we show that the XXZ type lambda model gives the symmetric space SU(2)/U(1) lambda model and, hence, we are able to find its spectrum and S-matrix and show that it gives the S-matrix of the O(3) sigma model in the appropriate limit. Finally, we show the full consistency of the S-matrix and the Lagrangian formulations of the lambda model, by coupling to a conserved charge and computing the way the ground state energy changes in both pictures.