Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections

TL;DR

This paper investigates the gauging of modulated symmetries in 1+1D spin chains, revealing conditions for duality isomorphisms and constructing non-invertible reflection symmetries, expanding understanding of dualities beyond traditional symmetries.

Contribution

It introduces a systematic framework for gauging finite Abelian modulated symmetries, establishing conditions for duality isomorphisms, and constructing non-invertible reflection symmetries in spin chains.

Findings

Duality isomorphisms are guaranteed by translation invariance for prime qudits.

Ring theory techniques show isomorphisms can exist for non-prime qudits without translation symmetry.

Non-invertible reflection symmetries can be constructed even without ordinary reflection symmetry.

Abstract

Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.