Global symmetries of quantum lattice models under non-invertible dualities

TL;DR

This paper investigates how global symmetries transform under non-invertible dualities in (1+1)-dimensional quantum lattice models, revealing algebraic structures and providing concrete examples with XXZ models.

Contribution

It introduces a framework for understanding global symmetry transformations under non-invertible dualities, including conjectures and examples in quantum lattice models.

Findings

Global symmetries form an algebraic ring of double cosets.

Concrete examples with XXZ models support the conjectures.

Provides a method to determine dual model symmetries from Hilbert space sectors.

Abstract

Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group $G$ in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the $\mathrm{Rep}(G)$ symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.