Haldane limits via Lagrangian embeddings

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.

29 pages, 2 figures

References in corpus (5)

• Comments on operators with large spin
• The Sigma Model on Complex Projective Superspaces
• Spontaneous trimerization in a bilinear-biquadratic S=1 zig-zag chain
• The worldsheet low-energy limit of the AdS_4 x CP^3 superstring
• Exact models for trimerization and tetramerization in spin chains

Cited by in corpus (27)

• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Generalization of the Haldane conjecture to SU(3) chains
• Fractional $\theta$ angle, 't Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model
• Dual simulation of the 2d U(1) gauge Higgs model at topological angle $\theta =\pi$: Critical endpoint behavior
• Sigma Models on Flags
• Integrable properties of sigma-models with non-symmetric target spaces
• The geometry of antiferromagnetic spin chains
• Flag manifold sigma models: spin chains and integrable theories
• Generalization of the Haldane conjecture to SU($n$) chains
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• First-principle simulations of 1+1d quantum field theories at $\theta =\pi$ and spin-chains
• Flag manifold sigma-models: the 1/N-expansion and the anomaly two-form
• Anomalies and unnatural stability of multi-component Luttinger liquids in ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ spin chains
• Topological terms of (2+1)d flag-manifold sigma models
• Mass generation by fractional instantons in SU($n$) chains
• $SU(3)$ Knot Solitons: Hopfions in the $F_2$ Skyrme-Faddeev-Niemi model
• Self-conjugate representation SU(3) chains
• Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach
• BPS Sphalerons in the $F_2$ Non-Linear Sigma Model
• An Index for Quantum Integrability
• Flag manifold sigma models from SU($n$) chains
• The classical and quantum particle on a flag manifold
• Winding theta and destructive interference of instantons
• Even-odd effects in the $J_1-J_2$ SU($N$) Heisenberg spin chain
• Grassmannian and Flag sigma models on interval: phase structure and L-dependence
• Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice
• Oscillator Calculus on Coadjoint Orbits and Index Theorems