Competing topological orders in three dimensions: X-cube versus toric code

TL;DR

This paper investigates the competition between X-cube and toric code topological orders in three dimensions, revealing new phases and characterizing phase transitions as first order through series expansions.

Contribution

It introduces a detailed analysis of the phase diagram for 3D topological orders, discovering two new phases connected to classical limits with degeneracies.

Findings

Identified two new phases with nontrivial degeneracies.

All phase transitions are first order.

Provided high-order series expansions of ground-state energies.

Abstract

We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displays a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.