Topological Quantum Phase Transitions in Topological Superconductors

TL;DR

This paper explores topological phase transitions in BF topological superconductors, revealing how different topologically ordered phases emerge through defect condensation, with a focus on (2+1)D models and their ground state degeneracies.

Contribution

It demonstrates that BF topological superconductors exhibit phase transitions between distinct topological phases characterized by different ground state degeneracies, especially in the (2+1)D case with compact gauge fields.

Findings

Superconducting phase has a ground state degeneracy of k.

Phase transitions are driven by topological defect condensation.

Compact gauge fields prevent certain phase configurations dynamically.

Abstract

In this paper we show that BF topological superconductors (insulators) exibit phase transitions between different topologically ordered phases characterized by different ground state degeneracy on manifold with non-trivial topology. These phase transitions are induced by the condensation (or lack of) of topological defects. We concentrate on the (2+1)-dimensional case where the BF model reduce to a mixed Chern-Simons term and we show that the superconducting phase has a ground state degeneracy $k$ and not $k^2$. When the symmetry is $U(1) \times U(1)$, namely when both gauge fields are compact, this model is not equivalent to the sum of two Chern-Simons term with opposite chirality, even if naively diagonalizable. This is due to the fact that U(1) symmetry requires an ultraviolet regularization that make the diagonalization impossible. This can be clearly seen using a lattice regularization, where the gauge fields become angular variables. Moreover we will show that the phase in which both gauge fields are compact is not allowed dynamically.