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

Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, Ömer M. Aksoy

TL;DR

This work develops a Hamiltonian framework for gauging finite Abelian modulated symmetries in 1+1D lattice models and uncovers a lattice-reflection–mediated isomorphism between original and gauged symmetries. It reveals that gauging modulated symmetries yields Kramers-Wannier dualities realized as non-invertible reflection operators, with explicit constructions in prime and general N cases via bond-algebra methods and ring-theoretic tools. The approach introduces sequential gauging where direct gauging fails, analyzes self-duality conditions across Z_N multipole families, and provides explicit KW operators for various modulated symmetries. The results illuminate the rich structure of dualities and non-invertible symmetries in modulated systems, with potential implications for symmetry defects, topological holography, and higher-dimensional generalizations.

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.

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

TL;DR

This work develops a Hamiltonian framework for gauging finite Abelian modulated symmetries in 1+1D lattice models and uncovers a lattice-reflection–mediated isomorphism between original and gauged symmetries. It reveals that gauging modulated symmetries yields Kramers-Wannier dualities realized as non-invertible reflection operators, with explicit constructions in prime and general N cases via bond-algebra methods and ring-theoretic tools. The approach introduces sequential gauging where direct gauging fails, analyzes self-duality conditions across Z_N multipole families, and provides explicit KW operators for various modulated symmetries. The results illuminate the rich structure of dualities and non-invertible symmetries in modulated systems, with potential implications for symmetry defects, topological holography, and higher-dimensional generalizations.

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 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.
Paper Structure (38 sections, 2 theorems, 345 equations)

This paper contains 38 sections, 2 theorems, 345 equations.

Table of Contents

  1. Introduction
  2. Modulated symmetries
  3. Summary
  4. Finite Abelian modulated symmetries from prime qudits
  5. Some simple examples
  6. Exponential symmetries
  7. Sums of exponentials symmetry and trigonometric symmetry
  8. Polynomial symmetries
  9. General case
  10. The bond algebra
  11. Gauging
  12. Dual symmetry
  13. General finite Abelian modulated symmetries
  14. Aspects of a general theory
  15. Examples
  16. ...and 23 more sections

Key Result

Theorem 1

An ${n\times r}$ matrix ${\cal F}$ over the commutative ring ${R}$ with determinantal rank ${t := \rho( {\cal F} )}$ is regular if and only if there exist idempotents ${e_0,\ldots, e_{t}\in E(R)}$ such that

Theorems & Definitions (2)

  • Theorem 1: Decomposition theorem
  • Theorem 2: Rank-nullity theorem