The Renormalization Group for Flag Manifolds

TL;DR

This paper derives renormalization group equations for quantum sigma-models on flag manifolds, analyzing fixed points and symmetry enhancements, with implications for antiferromagnetic systems and relativistic limits.

Contribution

It introduces RG equations for sigma-models on flag manifolds and explores fixed points and symmetry enlargements, extending understanding of these models in quantum field theory.

Findings

Existence of a fixed point at zero temperature in (2+ε)-dimensions.

Half of the couplings are irrelevant near the fixed point.

Global isometry groups enlarge at the fixed point.

Abstract

The renormalization group equations for a class of non--relativistic quantum $\sigma$--models targeted on flag manifolds are given. These models emerge in a continuum limit of generalized Heisenberg antiferromagnets. The case of the ${SU(3)\over U(1)\times U(1)}$ manifold is studied in greater detail. We show that at zero temperature there is a fixed point of the RG transformations in $(2+\varepsilon )$--dimensions where the theory becomes relativistic. We study the linearized RG transformations in the vicinity of this fixed point and show that half of the couplings are irrelevant. We also show that at this fixed point there is an enlargement of the global isometries of the target manifold. We construct a discrete non--abelian enlargement of this kind.