Commuting projector models for ( 3+1 )-dimensional topological superconductors via a string net of ( 1+1 )-dimensional topological superconductors

Abstract

We discuss a way to construct a commuting projector Hamiltonian model for a ($3+1$)-dimensional topological superconductor in class DIII. The wave function is given by a sort of string net of the Kitaev wire, decorated on the time-reversal ($T$) domain wall. Our Hamiltonian is provided on a generic three-dimensional (3D) manifold equipped with a discrete form of the spin structure. We will see how the 3D spin structure induces a 2D spin structure (called a ``Kasteleyn'' direction on a 2D lattice) on $T$ domain walls, which makes it possible to define fluctuating Kitaev wires on them. Upon breaking the $T$ symmetry in our model, we find the unbroken remnant of the symmetry, which is defined on the time-reversal domain wall. The domain wall supports the 2D nontrivial symmetry-protected topological (SPT) phase protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent quantum field theory argument by Hason, Komargodski, and Thorngren.