Haldane limits via Lagrangian embeddings
Dmitri Bykov
TL;DR
The paper addresses how certain $SU(N+1)$ spin chains admit continuum limits that are relativistic sigma models with target spaces given by flag manifolds $\mathcal{F}_{N+1}$. Using coherent-state path integrals and a detailed treatment of ferromagnetic and antiferromagnetic limits, the authors show that the antiferromagnetic sector for the $SU(3)$ case yields a flag sigma model with a normal metric, obtained by an isometric and Lagrangian embedding of $\mathcal{F}_3$ into $(\mathbf{C}\mathrm{P}^{2})^{\times 3}$; the theta-term is constrained by the embedding, effectively vanishing in this setup. They generalize to arbitrary $N$, demonstrating a continuum theory on $\mathcal{F}_{N+1}$ with an $S_{N+1}$-symmetric metric, and discuss the mass gap and connections to trimerization ($N$-merization). The work provides a systematic framework linking Haldane limits to Lagrangian embeddings and suggests broader applications to other isotropic embeddings and potential integrability properties. The results have potential relevance for condensed-matter systems exhibiting multi-site bound states and for AdS/CFT-inspired spin-foam-type constructions where flag manifolds play a role.
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.