Microscopic study of topological phase transitions: Percolation point of view
Phys. Rev. B 114, 214109 – Published 23 October, 2026
DOI: https://doi.org/10.1103/1rkh-nhq5
Abstract
In this paper, we investigate how a topologically ordered state changes to another one from a microscopic point of view. To this end, we focus on the behavior of topological entanglement and 1-form symmetry under decoherence, which induces a phase transition between these states. After a detailed numerical study of the color code as a first case study, we explain how the decoherence changes the topological properties of the state, observing topological entanglement negativity (TEN) and the disorder parameter of the 1-form symmetry. Our results suggest that the above quantities are closely connected with the first and 0th simplicial homology, respectively, and that the transition can be understood from the point of view of decoherence percolation. For a detailed microscopic investigation, we introduce the concept of quasilocal TEN (QLTEN) and show snapshots of the local topological property of the state, which is capable of clearly describing the decoherence percolation. To further investigate how QLTEN and the 1-form symmetry are related to the transition, we introduce the concept of explosive percolation (EP) to the present system, which has received a lot of attention in the last two decades. EP in the present context means that there exists some bias in the decoherence, reducing a large cluster of decohered regions. As a second case study, we consider a toric code on a triangular lattice under an external-field-type decoherence. Detailed numerical study reveals that QLTEN can faithfully describe the emergent Higgs region and the surviving logical qubits. We find that although topological entanglement negativity (TEN), clusters of QLTEN, and string operators strongly correlate with each other in both the color code and the toric code, the two systems react to patterns of EP quite differently. This fact indicates that obtaining a universal understanding of the topological phase transition is challenging even for strongly related systems.
