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arXiv:2504.03512 · PRX Quantum

Independent e- and m-Anyon Confinement in the Parallel Field Toric Code on Non-Square Lattices

Simon M. Linsel, Lode Pollet, Fabian Grusdt

Published Dec 5, 2025• 3 citations
arXivHTMLPDF

Abstract

Kitaev’s toric code has become one of the most studied models in physics and is highly relevant to the fields of both quantum error correction and condensed matter physics. Most notably, it is the simplest known model hosting an extended, deconfined topological bulk phase. To this day, it remains challenging to reliably and robustly probe topological phases, as many state-of-the-art order parameters are sensitive to specific models and even specific parameter regimes. With the emergence of powerful quantum simulators which are approaching the regimes of topological bulk phases, there is a timely need for experimentally accessible order parameters. Here we study the ground state physics of the parallel field toric code on the honeycomb, triangular, and cubic lattices using continuous-time quantum Monte Carlo. By extending the concept of experimentally accessible percolation-inspired order parameters (POPs) we show that electric and magnetic anyons are independently confined on the honeycomb and triangular lattices, unlike on the square lattice. Our work manifestly demonstrates that, even in the ground state, we must make a distinction between topological order and (de-)confinement. Moreover, we report multicritical points in the aforementioned confinement phase diagrams. Finally, we map out the topological phase diagrams on the honeycomb, triangular, and cubic lattices and compare the performance of the POPs with other topological order parameters. Our work paves the way for studies of confinement involving dynamical matter and the associated multicritical points in contemporary quantum simulation platforms for Z 2 lattice gauge theories.

DOI
10.1103/gtth-cclr

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Abstract The Kitaev honeycomb model is an approximate topological quantum error correcting code in the same phase as the toric code, but requiring only a 2-body Hamiltonian. As a frustrated spin model, it is well outside the commuting models of topological quantum codes that are typically studied, but its exact solubility makes it more amenable to analysis of effects arising in this noncommutative setting than a generic topologically ordered Hamiltonian. Here we study quantum error correction in the honeycomb model using both analytic and numerical techniques. We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space. These bounds are tighter than can be achieved using known general properties of topological phases. Our proofs are specialized to the honeycomb model, but some of the methods may nonetheless be of broader interest. Following this, we numerically study noise caused by thermalization processes in the perturbative regime close to the toric code renormalization group fixed point. The appearance of non-topological excitations in this setting has no significant effect on the error correction properties of the honeycomb model in the regimes we study. Although the behavior of this model is found to be qualitatively similar to that of the standard toric code in most regimes, we find numerical evidence of an interesting effect in the low-temperature, finite-size regime where a preferred lattice direction emerges and anyon diffusion is geometrically constrained. We expect this effect to yield an improvement in the scaling of the lifetime with system size as compared to the standard toric code.