Symmetric higher rank topological phases on generic graphs

TL;DR

This paper investigates the properties of 2D $ ext{Z}_N$ topological phases with fractional excitations on arbitrary graph-based lattices, deriving ground state degeneracy formulas linked to graph Laplacian invariants.

Contribution

It introduces a novel geometric framework using graph theory to analyze topological phases and derives a formula for ground state degeneracy based on graph Laplacian invariants.

Findings

Ground state degeneracy depends on invariant factors of the Laplacian.

Fractional excitations exhibit restricted mobility influenced by lattice geometry.

The framework generalizes topological phase analysis to arbitrary graph structures.

Abstract

Motivated by recent interests in fracton topological phases, we explore the interplay between gapped 2D $\mathbb{Z}_N$ topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a new geometric context -- graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, $G_x\boxtimes G_y$, we study the behavior of fractional excitations of the phases. We derive the formula of the ground state degeneracy of the phases, which depends on invariant factors of the Laplacian.