Commuting projector models for (3+1)d topological superconductors via string net of (1+1)d topological superconductors

TL;DR

This paper constructs a commuting projector Hamiltonian for a (3+1)d topological superconductor using a string net of Kitaev wires decorated on time reversal domain walls, revealing new insights into its topological classification.

Contribution

It introduces a novel 3D commuting projector model for class DIII topological superconductors using decorated string nets on domain walls, incorporating spin structures and symmetry considerations.

Findings

The model is defined on a 3D manifold with a discrete spin structure.

Time reversal domain walls support fluctuating Kitaev wires with non-trivial SPT properties.

The topological classification aligns with recent quantum field theory predictions.

Abstract

We discuss a way to construct a commuting projector Hamiltonian model for a (3+1)d 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 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 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 non-trivial SPT protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent QFT argument by Hason, Komargodski, and Thorngren.