# Quantum robustness and phase transitions of the 3D Toric Code in a field

## TL;DR

This paper investigates the stability of 3D topological order in the toric code model under magnetic fields, identifying phase transitions and the role of quasiparticle mobility in topological breakdown.

## Contribution

It introduces a variational scheme to analyze phase transitions in the 3D toric code under arbitrary magnetic fields, highlighting the importance of quasiparticle mobility.

## Key findings

- Identified first and second-order quantum phase transitions.
- Developed an approximate variational method for general field directions.
- Found that quasiparticle mobility influences topological order breakdown.

## Abstract

We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics.