Self-dual $S_3$ gauge theory in 2+1d: lattice model and topological phase transitions

Abstract:Electric-magnetic self-duality of the $\mathbb{Z}_2$ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the $S_3$ quantum double $\mathcal{D}(S_3)$ on a tensor product Hilbert space in which the $\mathbb{Z}^{\mathrm{em}}_2$ anyon-permutation symmetry, exchanging the non-Abelian chargeon $C$ and fluxon $F$, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent $\mathbb{Z}_2^{\mathrm{em}}$-preserving bosonic perturbations, driving $\mathcal{D}(S_3)$ into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles $\mathcal{D}(D_{2n})$. Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.

Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)

arXiv:2608.05294 [cond-mat.str-el]

(or arXiv:2608.05294v1 [cond-mat.str-el] for this version)

https://doi.org/10.48550/arXiv.2608.05294 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.