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Illustrating the Categorical Landau Paradigm in Lattice Models

Lakshya Bhardwaj, Lea E. Bottini, Sakura Schafer-Nameki, Apoorv Tiwari

Abstract

Recent years have seen the concept of global symmetry extended to non-invertible (or categorical) symmetries, for which composition of symmetry generators is not necessarily invertible. Such non-invertible symmetries lead to a generalization of the standard Landau paradigm. In this work we substantiate this framework by providing a (1+1)d lattice model, whose gapped phases and phase transitions can only be explained by symmetry breaking of non-invertible symmetries.

Illustrating the Categorical Landau Paradigm in Lattice Models

Abstract

Recent years have seen the concept of global symmetry extended to non-invertible (or categorical) symmetries, for which composition of symmetry generators is not necessarily invertible. Such non-invertible symmetries lead to a generalization of the standard Landau paradigm. In this work we substantiate this framework by providing a (1+1)d lattice model, whose gapped phases and phase transitions can only be explained by symmetry breaking of non-invertible symmetries.
Paper Structure (3 sections, 57 equations, 1 figure)

This paper contains 3 sections, 57 equations, 1 figure.

Table of Contents

  1. Ground States of the Hamiltonian $H_4$
  2. Proof that $\mathsf{Rep}(S_3)$ Symmetries Commute with the Hamiltonian
  3. Action of $E$ in the igSSB-Phase

Figures (1)

  • Figure 1: The Hamiltonian (\ref{['eq:Ham_3site']}) has four gapped phases. These are explained by the $\mathsf{Rep}(S_3)$ non-invertible symmetry breaking, whose action on the gapped ground states GS and gapless states is shown in blue (for symmetry operator $U$) and red (for symmetry operator $E$), with purple showing the full $\mathsf{Rep}(S_3)$ action. The phase transitions are indicated by black arrows. Non-zero vevs of order parameters are shaded yellow (for $O_q$) and blue (for $O_p$), with their intersection region shaded green.