Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model

TL;DR

This paper investigates the phase transition in a deformed toric code model, revealing an unconventional confinement mechanism and the persistence of ground state degeneracy linked to 1-form symmetry breaking, with implications for topological phases.

Contribution

It uncovers a novel confinement mechanism and demonstrates the persistence of ground state degeneracy across the phase transition, connecting topological order with 1-form symmetry breaking.

Findings

Unconventional confinement mechanism in trivial phase

Magnetic charges can be displaced without energy cost via non-unitary operators

Ground state degeneracy persists throughout the phase transition

Abstract

We examine the condensation and confinement mechanisms exhibited by a deformed toric code model proposed in [Castelnovo and Chamon, Phys. Rev. B, 2008]. The model describes both sides of a phase transition from a topological phase to a trivial phase. Our findings reveal an unconventional confinement mechanism that governs the behavior of the toric code excitations within the trivial phase. Specifically, the confined magnetic charge can still be displaced without any energy cost, albeit only via the application of non-unitary operators that reduce the norm of the state. This peculiar phenomenon can be attributed to a previously known feature of the model: it maintains the non-trivial ground state degeneracy of the toric code throughout the transition. We describe how this degeneracy arises in both phases in terms of spontaneous symmetry breaking of a generalized (1-form) symmetry and explain why such symmetry breaking is compatible with the trivial phase. The present study implies the existence of subtle considerations that must be addressed in the context of recently posited connections between topological phases and broken higher-form symmetries.