Haldane limits via Lagrangian embeddings

TL;DR

This paper explores the Haldane limit in spin chains, revealing its connection to Lagrangian embeddings, and constructs a specific spin chain limit leading to a relativistic sigma model on flag manifolds.

Contribution

It demonstrates the relation between the Haldane limit and Lagrangian embeddings, and constructs a new spin chain model with a specific continuum limit to a relativistic sigma model.

Findings

Haldane limit is related to Lagrangian embeddings in phase space.

Constructed a spin chain whose continuum limit yields a relativistic sigma model.

Identified the target space as the manifold of complete flags U(N)/U(1)^N.

Abstract

In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(N)/U(1)^N. We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.