Microscopic study of topological phase transitions: Percolation point of view

AI summary

Overview

Research area: Quantum physics — topological order, quantum entanglement, and decoherence-induced phase transitions, studied through percolation theory and simplicial homology.

Technical level: Advanced. The paper assumes familiarity with the stabilizer formalism, toric and color codes, entanglement negativity, and 1-form symmetries.

Scope (one sentence): A microscopic, numerical study of how decoherence drives a topologically ordered state into another topological state (color code to toric code) or into a non-topological one (toric code to Higgs phase), interpreted through percolation, network growth, and simplicial homology.

What This Paper Is About

Topologically ordered states have two strongly correlated aspects — quantum entanglement and 1-form symmetry — but it is not fully understood how the two are intertwined in the critical regime of a topological phase transition, or whether their indicators become singular at the same point with the same criticality. The authors address this by tracking a color code under red-link XX decoherence (and, as a second case, a toric code on a triangular lattice under external-field-type decoherence), extracting a "quasi-local" version of topological entanglement negativity so that the transition can be watched locally as a percolation process. They then ask how robust these topological states are when the decoherence pattern itself is biased.

Key Contributions

1. A quasi-local topological entanglement negativity (QLTEN). The authors introduce a microscopic analogue of TEN, framed as a cohomology of the (quasi-)closed sequence of S^Z-plaquette stabilizers, intended to be the 1st-homology counterpart of the disorder parameter (which they argue measures the 0th homology). Snapshots of QLTEN over the whole lattice give a geometric picture of the transition.

2. A percolation and simplicial-homology reading of the color-code-to-toric-code transition. They show analytically and numerically how XX decoherence on red links acts as network growth on a triangular lattice of red plaquettes, merging S^Z plaquette stabilizers, and they identify which decoherence configurations reduce TEN from 2 to 1.

3. Explosive percolation (EP) as a decoherence pattern. They import the concept of EP — biasing the decoherence rule so that large clusters of decohered regions are suppressed — into the topological setting, and test how TEN, QLTEN clusters, and string operators respond.

4. A second case study comparing two related systems. For the toric code on a triangular lattice under external-field-type decoherence, QLTEN is shown to describe the emergent Higgs region and the surviving logical qubits; comparing the two systems under the same EP rules reveals a qualitative difference in response.

Main Findings

• The color code does convert to a toric code. At decoherence probability p = 1, the red-link XX decoherence leaves link qubits with operators Ž = Z_{v1}Z_{v2} and X̌ = X_{v1} or X_{v2}, and the color code stabilizers map onto toric-code stabilizers on the emergent red triangular lattice, while the other toric-code factor is driven to a trivial state by X-anyon proliferation.

• TEN drops from 2 to 1. Figure 5 shows TEN changing from the color code value γ_N = 2 to the toric code value γ_N = 1, with clear subsystem-size dependence; the variance of TEN has a large, smooth peak at p ≃ 0.22, which is noticeably different from the bond percolation threshold on the triangular lattice, 0.347.

• The 1-form symmetry signals a sharper transition. The disorder parameter D^X(Γ) rises from 0 to 1 with increasing p, and its variance gives a continuous transition with p_s ≃ 0.31 and ν ≃ 1.2 from finite-size scaling, values the authors describe as fairly close to the bond percolation point of the triangular lattice. Data shown use N_s ~ 10^3 decoherence realizations and system size (L_x, L_y) = (48, 32).

• The discrepancy has a homology explanation. The disorder parameter detects a connected 0th-complex cluster of S^Z plaquette stabilizers linking the two string endpoints, whereas TEN requires a (quasi-)closed 1st-complex loop; the authors argue this dimensional difference is a candidate origin of the numerically distinct behaviors.

• A single decoherence event costs exactly one unit of negativity. For a 7-plaquette hexagonal subsystem (|∂A| = 12), one red XX decoherence lowers rank K̃_{2A} by 2 and N_A by 1; after ½|∂A| − 1 such operations all boundary S^Z plaquettes have merged into one large stabilizer and further decoherence is redundant, consistent with K̃_{2A} − 2(½|∂A| − 1) = 0.

• TEN = 1 requires a (quasi-)closed subsystem. Configurations in which at least one of the three subsystems A, B, C is (quasi-)closed — all boundary Z-plaquette operators connected by decoherence — yield γ_N = 1, while the K̃_1 contribution is unaffected.

• Snapshots show full conversion by p = 0.4. QLTEN snapshots (displayed on a honeycomb-to-square lattice map) show the whole system in the toric-code region at p = 0.4, consistent with the mean TEN.

• The two systems react differently to explosive percolation. TEN, QLTEN clusters, and string operators strongly correlate with each other in both the color code and the toric code, yet the two systems respond quite differently to EP patterns — leading the authors to conclude that a universal understanding of the topological phase transition is challenging even for strongly related systems.

Methodology in Plain English

The authors work entirely within the stabilizer formalism, where a state is encoded by a set of commuting Pauli strings and every update to the state is tracked by how that set changes. They start from the color-code ground state on a honeycomb lattice and apply a stochastic quantum channel: each red link independently receives an XX operator with probability p, and when it does the channel parameter is set to p_r = 1/2 (decoherence, with no measurement record kept, so the state becomes mixed). Because each XX merges two S^Z plaquette stabilizers, this is literally a random network-growth process on the triangular lattice whose vertices are red plaquettes. For each sampled configuration they compute two observables: the topological entanglement negativity, obtained from the rank over the binary field of a matrix built from truncated stabilizer generators for three adjacent subsystems A, B, C, and the disorder parameter of the 1-form symmetry, obtained from a string operator whose two endpoints must land in one connected stabilizer cluster. Averages over roughly a thousand realizations produce the curves, and finite-size scaling on the variance locates the critical point. They then repeat the analysis with modified growth rules that suppress large clusters, which is the explosive-percolation variant, and finally apply the same toolkit to a toric code on a triangular lattice under external-field-type decoherence, watching where the Higgs region forms and which logical qubits survive.

Why This Matters

Impact on research. The paper supplies a concrete, quantitative link between two usually separate diagnostics of topological order — entanglement negativity and 1-form symmetry — and shows they need not go critical together. Framing the transition as percolation plus simplicial homology gives a microscopic mechanism (cluster growth versus closed-loop formation) rather than a purely phenomenological description, and the finding that two closely related models respond differently to biased decoherence cautions against assuming universality across topological phase transitions.

Real-world applications (as framed by the paper's motivation):

• Quantum error mitigation and fault tolerance, since the survival of logical qubits is tied to QLTEN cluster formation in the Higgs region.
• Designing decoders and error-correction thresholds for color-code and toric-code architectures.
• Predicting how biased or correlated noise (as opposed to uniform noise) degrades a topological quantum memory.
• Benchmarking topological order in mixed states produced by real devices, where decoherence rather than measurement is the dominant effect.

Industry relevance. Any hardware platform pursuing topological quantum computing — superconducting circuits, trapped ions, neutral atoms — must characterize how its noise profile interacts with topological protection; the paper's point that the pattern of decoherence matters, not just its strength, is directly relevant to noise-engineering and calibration strategies.

Future Directions

• Establish whether QLTEN can be made a quantitative order parameter with its own scaling law and critical exponents, beyond the qualitative snapshots presented.
• Resolve the mismatch in critical points between TEN (variance peak near p ≃ 0.22) and the 1-form symmetry (p_s ≃ 0.31), for example by testing whether the large TEN variance originates from non-local correlations between subsystem negativities, as the authors speculate.
• Extend the EP analysis to further models and other bias rules, given that the color code and the toric code responded differently, to determine which features of a decoherence pattern generically control the transition.
• Connect the QLTEN cluster picture to practical decoding and error mitigation, since the Higgs-region clusters are argued to be the key to fault tolerance of the toric code.

Target Audience

Researchers and graduate students in quantum information and condensed-matter physics working on topological order, decoherence-induced phase transitions, or entanglement measures in mixed states. It will be most useful to readers already comfortable with the stabilizer formalism and toric/color codes; readers from classical percolation theory and network science may also find the explosive-percolation adaptation of interest.

Authors’ 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.