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arXiv:2607.26707 · Physical review. B./Physical review. B

Microscopic study of topological phase transitions: Percolation point of view

Keisuke Kataoka, Ikuo Ichinose

quant-phcond-mat.othercond-mat.stat-mechcond-mat.str-elSubmitted Jul 29, 2026• Announced Jul 30, 2026• 0 citations• Revised Aug 2, 2026 (v2)
v1 · Jul 29, 2026v2 · Aug 2, 2026v3 · Oct 21, 2026
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Abstract

We investigate microscopic mechanisms underlying decoherence-induced transitions between topologically ordered states. As a first case study, we analyze the color code using topological entanglement negativity (TEN) and a disorder parameter associated with 1-form symmetry. We interpret these quantities as first- and zeroth-dimensional simplicial homological objects, respectively, and show that the transition can be understood in terms of decoherence percolation. To resolve its local structure, we introduce quasi-local TEN (QLTEN), which visualizes the spatial distribution of local topological properties and the growth of decohered regions. We further introduce explosive percolation (EP), corresponding here to biased decoherence that suppresses the formation of large decohered clusters. As a second case study, we consider the toric code on a triangular lattice under external-field-type decoherence. Numerical results show that QLTEN faithfully captures the emergence of Higgs regions and the survival of logical qubits. Although global TEN, QLTEN clusters, and string operators are strongly correlated in both models, the color code and toric code respond differently to EP patterns. This difference indicates that a universal microscopic description of decoherence-induced topological phase transitions remains challenging even for closely related systems.

Journal reference
Phys. Rev. B 114, 214109 (2026)
DOI
10.1103/1rkh-nhq5
Comments
21 pages, 19 main figures and 2 figures in Appendix. Revised version submitted to Physical Review B

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We investigate microscopic mechanisms underlying decoherence-induced transitions between topologically ordered states. As a first case study, we analyze the color code using topological entanglement negativity (TEN) and a disorder parameter associated with 1-form symmetry. We interpret these quantities as first- and zeroth-dimensional simplicial homological objects, respectively, and show that the transition can be understood in terms of decoherence percolation. To resolve its local structure, we introduce quasi-local TEN (QLTEN), which visualizes the spatial distribution of local topological properties and the growth of decohered regions.
We further introduce explosive percolation (EP), corresponding here to biased decoherence that suppresses the formation of large decohered clusters. As a second case study, we consider the toric code on a triangular lattice under external-field-type decoherence. Numerical results show that QLTEN faithfully captures the emergence of Higgs regions and the survival of logical qubits. Although global TEN, QLTEN clusters, and string operators are strongly correlated in both models, the color code and toric code respond differently to EP patterns. This difference indicates that a universal microscopic description of decoherence-induced topological phase transitions remains challenging even for closely related systems.
We study how the color code under decoherence gives rise to an intrinsic mixed-state topological order (imTO), which has no counterpart in pure ground states of local gapped Hamiltonians. For decoherence induced by XX-type operators on red links of the honeycomb lattice, we show that the resulting mixed state inherits half of the topological properties of the color code, including anyon content, logical operators, and topological entanglement structure. Using a gauging procedure for mixed stabilizer states, we identify the emergent phase as closely related to a single toric code. We characterize this phase by topological entanglement negativity (TEN) and perform efficient stabilizer-formalism simulations. While the pure color code has TEN=2ln⁡2{\rm TEN} = 2 \ln 2TEN=2ln2, the maximally decohered state has TEN=ln⁡2{\rm TEN} = \ln 2TEN=ln2, indicating emergence of a single toric code. By tuning the decoherence strength, we find a smooth crossover in TEN accompanied by a pronounced, nearly system-size-independent peak in its variance. We further show that the negativity exhibits characteristic scaling only for subsystem partitions commensurate with the triangular lattice of the emergent toric code. Our results demonstrate that negativity-based quantities provide powerful probes of mixed-state topological order generated by decoherence.
We study the entanglement structure of topological orders subject to decoherence on the bipartition boundary. Focusing on the toric codes in ddd space dimensions for d=2,3,4d=2,3,4d=2,3,4, we explore whether the boundary decoherence may be able to induce a disentangling transition, characterized by the destruction of mixed-state long-range entanglement across the bipartition, measured by topological entanglement negativity. A key insight of our approach is the connection between the negativity spectrum of the decohered mixed states and emergent symmetry-protected topological orders under certain symmetry-preserving perturbation localized on the bipartition boundary. This insight allows us to analytically derive the exact results of entanglement negativity and characterize the universality class of the disentangling transition without using a replica trick.
Decoherence on quantum many-body systems induces nontrivial mixed states with specific properties that have no counterparts in pure states. The authors clarify here the emergence of ``intrinsic mixed state topological order'' in a toric code state under decoherence, which accompanies exotic anyon proliferation. Furthermore, the authors find that the decoherence-induced phase transition in the stabilizer formalism is described by a percolation picture in two dimensions. The viewpoint of one-form symmetry and its spontaneous symmetry breaking plays an important role in this study.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.
Transition out of a topological phase is typically characterized by discontinuous changes in topological invariants along with bulk gap closings. However, as a clean system is geometrically punctured, it is natural to ask the fate of an underlying topological phase. To understand this physics we introduce and study both short- and long-ranged toy models where a one-dimensional topological phase is subjected to bond percolation protocols. We find that non-trivial boundary phenomena follow competing energy scales even while the global topological response is governed via geometrical properties of the percolated lattice. Using numerical, analytical, and appropriate mean-field studies we uncover the rich phenomenology and the various crossover regimes of these systems. In particular, we discuss the emergence of the ``fractured topological region'' where an overall trivial system contains a macroscopic number of topological clusters. Our study shows the interesting physics that can arise from an interplay of geometrical disorder within a topological phase.
Decoherence-induced phase transitions (DIPTs) are associated with singular changes in a quantum system's entanglement structure as the decoherence strength varies. In many cases, their critical strengths depend monotonically on the Rényi index of the entanglement diagnostic, forming a Rényi hierarchy. We identify a general mechanism for this hierarchy based on correlation inequalities in replicated statistical models that describe these DIPTs. We demonstrate it in Z2\mathbb{Z}_2Z2​ topological stabilizer codes subject to independent phase-flip decoherence on each qubit and in a decohered rotor model with strong U(1)\text{U}(1)U(1) symmetry. When the relevant correlations diagnose the DIPTs, these inequalities imply that the critical decoherence strength is nondecreasing with integer Rényi index R≥2R\geq2R≥2. We show that this mechanism also applies to a class of models with correlated decoherence.