The Renormalization Group Method and Quantum Groups: the postman always rings twice

TL;DR

This paper explores the connection between the Real-Space Renormalization Group method and Quantum Groups by analyzing quantum group invariant Hamiltonians, revealing a quantum group anomaly and deriving new RG equations that match known critical behavior.

Contribution

It introduces the concept of quantum group anomaly in RG transformations and derives new qRG equations for the XXZ model, linking RG flow to quantum group symmetries.

Findings

RG flow diagram matches the exact critical points

Quantum group anomaly appears in RG transformations

qRG equations align with tensor product decompositions

Abstract

We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearence of a quantum group anomalous term which vanishes in the classical case. This is called {\em the quantum group anomaly}. We derive the new qRG equations for the XXZ model and show that the RG-flow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. In the ITF model the qRG-flow equations coincide with the tensor product decomposition of cyclic irreps of $SU_q(2)$ with $q^4=1$.