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

Simon M. Linsel1,2,*, Lode Pollet1,2, and Fabian Grusdt1,2

  • *Contact author: simon.linsel@lmu.de
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PRX Quantum 7, 010332 – Published 13 February, 2026

DOI: https://doi.org/10.1103/gtth-cclr

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 Z2 lattice gauge theories.

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Popular Summary

Topological phases of matter feature exotic and intricate physics, including long-range entanglement and particlelike excitations (“anyons”) which are neither fermionic nor bosonic. Apart from being physically interesting, topological phases are also closely related to realizing error-corrected qubits (the analog of bits in a regular computer) which are required for harnessing the full potential of large-scale quantum computers in real-world scenarios where noise cannot be neglected. In this work, we shed light on the relation of the physical property of anyon confinement, measuring how freely anyons can move through the system, to the emergence of topological phases. We study a paradigmatic model hosting a topological ground state: the toric code subject to external fields, which—when strong enough—can destroy the topological order and alter the confinement of anyons.

We apply a numerical quantum Monte Carlo method to map out the confinement and topological phase diagrams on the honeycomb, triangular, and cubic lattices; these nonsquare geometries are predominantly realized in current-generation experiments exploring topological phases and thus highly relevant. We use special order parameters related to percolation theory, which are accessible to such experiments. We find that different kinds of anyons (electric and magnetic) can be independently confined, and the topological phase requires both types to be deconfined.

Studying anyon confinement in dynamical (i.e., time-evolving) systems or at finite temperature and its relation to topology is an interesting direction for future work.

Article Text

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PRX Quantum 7, 010332– Published 13 February, 2026
  • Received 10 April 2025
  • Revised 1 October 2025
  • Accepted 5 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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