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Odd toric code in a tilted field: Higgs-confinement multicriticality, spontaneous self-duality symmetry breaking, and valence bond solids

Umberto Borla1,2,3, Ayush De1, and Snir Gazit1,4

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Phys. Rev. B 113, 245134 – Published 16 June, 2026

DOI: https://doi.org/10.1103/wbvd-z9s4

Abstract

We investigate the quantum phase diagram of an “odd” variant of the two-dimensional Ising Fradkin-Shenker model, characterized by a uniform background of static e and m charges. Using large-scale tensor network and exact diagonalization methods, we determine the topology of the phase diagram, identifying an “odd” deconfined phase, confinement- and Higgs-induced valence bond solids (VBS), and a trivial paramagnet. Most notably, we uncover an exotic multicritical point along the self-dual line, where electric and magnetic excitations are related by an enriched Z2 duality. This transition is marked by the simultaneous onset of confinement, Higgs condensation, translational symmetry breaking, and spontaneous duality symmetry breaking. Within our numerical accuracy, the transition appears continuous, involving the softening of excitation gaps for e and m anyons at finite momentum. At intermediate couplings, we further identify VBS phases with enlarged unit cells, potentially indicating frustration-induced crystalline order beyond commensurate limits.

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Phys. Rev. B 113, 245134– Published 16 June, 2026
  • Received 5 August 2025
  • Revised 22 April 2026
  • Accepted 11 May 2026
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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. Open access publication funded by Max Planck Society.

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