Classification of subsystem symmetry-protected topological phases

TL;DR

This paper classifies 2D subsystem symmetry-protected topological phases, distinguishing strong from weak phases, and introduces a mathematical framework for their classification based on group cohomology.

Contribution

The paper proposes a natural definition of strong equivalence for subsystem SPT phases and classifies them using a novel cohomological group structure.

Findings

Strong subsystem SPT phases are classified by elements of a specific cohomology group.

Weak phases are shown to be equivalent to trivial phases under linearly-symmetric local unitaries.

Strong phases exhibit a spurious topological entanglement entropy on a cylinder.

Abstract

We consider symmetry-protected topological (SPT) phases in 2D protected by linear subsystem symmetries, i.e. those that act along rigid lines. There is a distinction between a "strong" subsystem SPT phase, and a "weak" one, which is composed of decoupled 1D SPTs with global symmetries. We propose a natural definition for strong equivalence of such phases, in terms of a linearly-symmetric local unitary transformation, under which a weak subsystem SPT is equivalent to the trivial phase. This leads to a number of distinct equivalence classes of strong subsystem SPTs, which we show are in one-to-one correspondence with elements of the group $\mathcal{C}[G_s] = \mathcal{H}^{2}[G_s^2,U(1)]/(\mathcal{H}^{2}[G_s,U(1)])^3$, where $G_s$ is the finite abelian onsite symmetry group. We also show that strong subsystem SPTs by our classification necessarily exhibit a spurious topological entanglement entropy on a cylinder.