Flag manifold sigma models: spin chains and integrable theories

Abstract

This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $C{P}^{n-1}$ and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete 't Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization of the theory of integrable models with symmetric target spaces. These models, as well as their deformations, have an alternative equivalent formulation as bosonic Gross-Neveu models, which proves useful for demonstrating that the deformed geometries are solutions of the renormalization group (Ricci flow) equations, as well as for the analysis of anomalies and for describing potential couplings to fermions.

146 pages, 22 figures

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• Integrable sigma models on Riemann surfaces
• Even-odd effects in the $J_1-J_2$ SU($N$) Heisenberg spin chain
• The Haldane gap in the SU(3) [3 0 0] Heisenberg chain
• Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum
• Lieb-Schultz-Mattis constraints for the insulating phases of the one-dimensional SU($N$) Kondo lattice model
• Oscillator Calculus on Coadjoint Orbits and Index Theorems
• $\beta$-function of the level-zero Gross-Neveu model