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The $\mathbb{Z}_N^{\times 3}$ symmetry protected boundary modes in two-dimensional Potts paramagnets

Hrant Topchyan

Abstract

We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with $\mathbb{Z}_N^{\times 3}$ symmetry on a triangular lattice. Using a cohomology-based transformation, the lattice models for the edge modes are explicitly obtained, and their structure is shown to be governed by the arithmetic properties of $N$. For prime $N$, the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of $N$, the models exhibit hierarchical or factorized structures. We demonstrate that all phases can be understood in terms of primary models augmented by local defect degrees of freedom that partition the system into independent segments. Finally, the global symmetry is realized on the boundary in a non-on-site and anomalous manner via a projective representation, directly realizing the corresponding 't Hooft anomaly.

The $\mathbb{Z}_N^{\times 3}$ symmetry protected boundary modes in two-dimensional Potts paramagnets

Abstract

We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with symmetry on a triangular lattice. Using a cohomology-based transformation, the lattice models for the edge modes are explicitly obtained, and their structure is shown to be governed by the arithmetic properties of . For prime , the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of , the models exhibit hierarchical or factorized structures. We demonstrate that all phases can be understood in terms of primary models augmented by local defect degrees of freedom that partition the system into independent segments. Finally, the global symmetry is realized on the boundary in a non-on-site and anomalous manner via a projective representation, directly realizing the corresponding 't Hooft anomaly.

Paper Structure

This paper contains 16 sections, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The considered system
  3. SPT phases
  4. The boundary modes
  5. Case study of different $N$-values
  6. Prime values
  7. Two-point representation
  8. Prime power values
  9. The primary phase
  10. The other phases
  11. General composite values
  12. Aspects of symmetries
  13. Winding and Laterality-like symmetries
  14. Number of active elements
  15. The 't Hooft anomaly
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The defined superlattice. Green, blue and yellow circles are the nodes of the original triangular lattice (shown in dark lines) with a $\mathbb{Z}_N$ degree of freedom. Red shaded groups are the supernodes with a $\mathbb{Z}_N^{\times3}$ degree of freedom, that define the triangular superlattice (red lines connecting the supernodes).
  • Figure 2: The visual representation of ${\cal W}_0$ for the case $f=5$. The axial lines representing different values of $n$ (thick dark lines) with the states $n_x$ at $x$ marked (green cubes) on them, and the polyline (yellow lines) connecting the markings. ${\cal W}_0$ is the number of revolutions of the polyline around the axis (light blue line).