A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph

TL;DR

This paper investigates a model where subsystem symmetry breaking and SPT order coexist, revealing how their mathematical structure leads to a highly degenerate ground state and dual order parameters.

Contribution

It introduces a model demonstrating the coexistence of subsystem symmetry breaking and SPT order, highlighting their mathematical origin and implications for ground state degeneracy.

Findings

Subsystem symmetries cause exponential ground state degeneracy.

Both Landau and SPT-like order parameters can be defined for each loop.

The model explains the coexistence of SSB and SPT orders mathematically.

Abstract

We analyze a model demonstrating the co-existence of subsystem symmetry breaking (SSB) and symmetry-protected topological (SPT) order, or subsystem LSPT order for short. Its mathematical origin is the existence of both a subsystem and a local operator, both of which commute with the Hamiltonian but anti-commute between themselves. The reason for the exponential growth of the ground state degeneracy is attributed to the existence of subsystem symmetries, which allows one to define both the Landau order parameter and the SPT-like order for each independent loop.