An Exactly Solvable Model for a $4+1D$ Beyond-Cohomology Symmetry Protected Topological Phase

TL;DR

This paper presents an exactly solvable model for a 4+1D beyond-cohomology SPT phase with unique boundary properties, including nontrivial symmetry actions and chiral edge states, expanding the understanding of higher-dimensional topological phases.

Contribution

It constructs a novel 4+1D exactly solvable model for a beyond-cohomology SPT phase with decorated domain walls and nontrivial boundary phenomena, including a nontrivial quantum cellular automaton.

Findings

The model describes a 4+1D beyond-cohomology SPT phase.

Boundary symmetry cannot be realized by a quantum circuit, but as a nontrivial QCA.

A gapped boundary with anomalous topological order is constructed.

Abstract

We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${\mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with "three-fermion" Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${\mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a ${\mathbb Z}_2$ domain wall at a trivial boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of arXiv:1602.00187, and define a corresponding anomaly indicator.