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  • Letter
  • Open Access

Continuous topological phase transition between Z2 topologically ordered phases

Qi Zhang1 and Wen-Tao Xu2,3,*

  • *Contact author: wen-tao.xu@tum.de
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Phys. Rev. B 112, L241107 – Published 8 December, 2025

DOI: https://doi.org/10.1103/1bjs-mf2l

Abstract

Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct Z2 topological orders in (2+1) dimensions [(2+1)D]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the Z4 topological order, we introduce a perturbed Z4 quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous (2+1)D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that Z4 anyons, fractionalized from the Z2 topological orders, become deconfined at the transition between Z2 topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.

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Phys. Rev. B 112, L241107– Published 8 December, 2025
  • Received 16 August 2025
  • Revised 30 October 2025
  • Accepted 12 November 2025
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