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  • Letter
  • Open Access

Fracton and topological order in the XY checkerboard toric code

Maximilian Vieweg and Kai Phillip Schmidt*

  • *Contact author: kai.phillip.schmidt@fau.de
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Phys. Rev. Research 7, L032048 – Published 3 September, 2025

DOI: https://doi.org/10.1103/msdd-ckq7

Abstract

We introduce the XY checkerboard toric code. It represents a generalization of the Z2 toric code with two types of star operators with x and y flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of a subextensive number of Z2 conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with Z2 topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase such as subextensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.

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Phys. Rev. Research 7, L032048– Published 3 September, 2025
  • Received 4 December 2024
  • Revised 5 March 2025
  • Accepted 23 June 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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