Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order

TL;DR

This paper constructs an exactly solvable 3D model of a bosonic symmetry protected topological phase with unique surface topological order, revealing new surface phenomena not realizable in 2D systems.

Contribution

It introduces a novel 3D exactly soluble Hamiltonian for a bosonic SPT phase with surface topological order, extending the Walker-Wang construction beyond known classifications.

Findings

Surface anyons match the 3-fermion Z2 model

The phase is outside the group cohomology classification

Surface exhibits a 'half' Kitaev E8 phase

Abstract

We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time reversal symmetry, i.e a symmetry protected topological (SPT) phase. In this model anyonic excitations are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the 3-fermion Z2 model, a variant of Z2 topological order which cannot be realized in a purely D=2 system with time reversal symmetry. Thus the model realizes a novel surface termination for SPT phases, which only becomes available in D=3: that of a fully symmetric gapped surface with topological order. The 3D phase found here is also outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. It is identified with a phase previously predicted from a field theoretic analysis, whose surface is roughly `half' of a Kitaev E8 phase. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons,which can be extended to other symmetry classes and topological orders.