Self-dual Higgs transitions: Toric code and beyond

TL;DR

The paper addresses the enigmatic continuous self-dual transition of the toric code by proposing a concrete continuum description, the $SO(4)_{2,-2}$ CS-Higgs theory, to capture both the toric-code phase with a self-duality and the topologically trivial phase with spontaneous $\,\mathbb{Z}_2\,$ symmetry breaking. It shows that this Higgs framework, involving an $SO(4)$ gauge field and a four-component real scalar, naturally yields the correct phase structure and can be generalized to the $SO(4)_{k,-k}$ family, describing transitions into various non-Abelian topological orders (e.g., $S_3$ quantum double for $k=4$, double Fibonacci for $k=3$). The work also explores a potential infrared-dual Ising description at $k=1$ and interprets the transitions in terms of condensations of unified non-Abelian anyons (e.g., $\, ext{"}\sigmaar{\sigma} ext{"} ight)$, connecting to particle-vortex dualities in related systems. A notable insight is the absence of additional IR symmetries at the multicritical point, which both informs the bootstrap landscape and motivates further numerical tests of the proposed framework and its large-$k$ expansions.

Abstract

The toric code, when deformed in a way that preserves the self-duality $\mathbb{Z}_2$ symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the $\mathbb{Z}_2$ symmetry. Numerically, this transition was found to be continuous, which makes it particularly enigmatic given the longstanding absence of a continuum field-theoretic description. In this work we propose such a continuum field theory for the transition dubbed the $SO(4)_{2,-2}$ Chern-Simons-Higgs (CSH) theory. We show that our field theory provides a natural "mean-field" understanding of the phase diagram. Moreover, it can be generalized to an entire series of theories, namely the $SO(4)_{k,-k}$ CSH theories, labeled by an integer $k$. For each $k>2$, the theory describes an analogous transition involving different non-Abelian topological orders, such as the double Fibonacci order ($k=3$) and the $S_3$ quantum double ($k=4$). For $k=1$, we conjecture that the corresponding CSH transition is in fact infrared-dual to the $3d$ Ising transition, in close analogy with the particle-vortex duality of a complex scalar.

Figures (1)

• Figure 1: Schematic phase diagram of the deformed toric code from numerical studies TupitsynKitaevProkofEvStampVidalDusuelSchmidtWuDengProkofevSomozaSernaNahumOppenheimKochJanuszGazitRingel. We are interested in the "multi-critical" point on the self-dual line $h_x=h_z$, where the two Ising$^*$ transitions meet, followed by a topologically trivial phase that spontaneously breaks the self-duality symmetry.