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A journey on self-$G$-ality

Takamasa Ando

TL;DR

The paper investigates self-$G$-ality in (1+1)D systems with fusion-category symmetries by employing SymTFT and topological manipulations. It builds a web of enhanced symmetry categories, notably TY and related constructions, through explicit analysis of $ ext{Z}_2 imes ext{Z}_2$ and $ ext{Z}_n imes ext{Z}_n$ models, and derives LSM-type constraints that forbid certain gapped phases under self-duality. It then connects these algebraic structures to concrete lattice realizations, mapping self-duality lines to XXZ/compact-boson CFTs and identifying KT points, orbifold branches, and mixed anomalies. The work introduces codimension-two transitions as Morita-equivalences between enhanced symmetry theories and demonstrates how dualities between self-dualities organize into a rich web, with implications for criticality, anomaly matching, and the classification of symmetry-enhanced phases. Overall, it lays a framework for understanding how higher-level symmetry data controls phase structure and critical behavior in low-dimensional quantum systems and suggests broad avenues for generalization and lattice-model construction.

Abstract

We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by the invariance of the actions of topological manipulations, i.e., self-$G$-alities for topological manipulations. Based on the self-$G$-ality conditions, we provide LSM-type constraints on the ground states of many-body Hamiltonians. We clarify the relationship between different enhanced symmetries and introduce the notion of $\textit{codimension-two transitions}$. We explore concrete lattice models for such self-$G$-alities and find how the self-$G$-ality structures match the IR critical theories.

A journey on self-$G$-ality

TL;DR

The paper investigates self--ality in (1+1)D systems with fusion-category symmetries by employing SymTFT and topological manipulations. It builds a web of enhanced symmetry categories, notably TY and related constructions, through explicit analysis of and models, and derives LSM-type constraints that forbid certain gapped phases under self-duality. It then connects these algebraic structures to concrete lattice realizations, mapping self-duality lines to XXZ/compact-boson CFTs and identifying KT points, orbifold branches, and mixed anomalies. The work introduces codimension-two transitions as Morita-equivalences between enhanced symmetry theories and demonstrates how dualities between self-dualities organize into a rich web, with implications for criticality, anomaly matching, and the classification of symmetry-enhanced phases. Overall, it lays a framework for understanding how higher-level symmetry data controls phase structure and critical behavior in low-dimensional quantum systems and suggests broad avenues for generalization and lattice-model construction.

Abstract

We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by the invariance of the actions of topological manipulations, i.e., self--alities for topological manipulations. Based on the self--ality conditions, we provide LSM-type constraints on the ground states of many-body Hamiltonians. We clarify the relationship between different enhanced symmetries and introduce the notion of . We explore concrete lattice models for such self--alities and find how the self--ality structures match the IR critical theories.
Paper Structure (19 sections, 2 theorems, 52 equations, 3 figures, 1 table)

This paper contains 19 sections, 2 theorems, 52 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Topological manipulation and Symmetry TFT
  3. Z2 x Z2 symmetry
  4. Gapped phases and topological manipulations
  5. Enhanced symmetry category
  6. No-go argument
  7. Dualities between self-dualities
  8. Codimension-two transition
  9. Lattice models for self-G-alities with Z2 x Z2 symmetry
  10. Self-duality lines
  11. Self-G-ality structures
  12. Zn x Zn symmetry
  13. SPT transition
  14. Outlook
  15. Field-theory analysis
  16. ...and 4 more sections

Key Result

Proposition 2

Let $p$ be an odd prime integer. There is a duality operation between any two $\mathbb{Z}_p\times\mathbb{Z}_p$ SPTs with different levels.

Figures (3)

  • Figure 1: The phase diagram of the Hamiltonian \ref{['Z2Z2_ham']}. Three red lines are self-duality lines, and the blue point in the center is self-$S_3$-ality point and it is described by the KT CFT.
  • Figure 2: The relation between the XXZ model and the Ashkin-Teller model.
  • Figure 3: Three gapped phases with $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry and the dualities between them. The group structure of the web is $SL(2,\mathbb{Z}_2)\cong S_3$, see Li:2023mmw.

Theorems & Definitions (3)

  • Claim 1
  • Proposition 2
  • Corollary 3