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

TL;DR

This paper explores an 'odd' variant of the 2D Ising Fradkin--Shenker model, revealing a rich phase diagram with exotic multicritical points, symmetry breaking, and valence bond solid phases using tensor network and exact diagonalization methods.

Contribution

It uncovers an exotic multicritical point with enriched duality and symmetry breaking in the 'odd' toric code model, expanding understanding of topological phase transitions.

Findings

Identification of an 'odd' deconfined phase and VBS phases.

Discovery of a multicritical point with simultaneous symmetry breakings.

Observation of continuous transition involving anyon gap softening.

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 $\mathbb{Z}_2$ 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.