The toric code, when deformed in a way that preserves the self-duality Z_{2} symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the 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 Letter 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.
The toric code, when deformed in a way that preserves the self-duality Z_{2} symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the 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 Letter 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.
Self-dual S3 gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2 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 S3 quantum double D(S3) on a tensor product Hilbert space in which the Z2em 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 Z2em-preserving bosonic perturbations, driving D(S3) 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 D(D2n). 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.
Self-dual criticality in three-dimensional Z2 gauge theory with matter
The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``e'' and ``m'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the Z2 gauge-Higgs model on cubic lattices of linear size L≤96. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators A and S. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions xA and xS and dynamical exponent z. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.
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 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.
From QED3 to Self-Dual Multicriticality in the Fradkin-Shenker Model
We consider the Fradkin-Shenker Z2 gauge-Higgs lattice model in 2+1 dimensions, i.e. the toric code deformed by an in-plane magnetic field. Its phase diagram contains a multicritical CFT with gapless, mutually non-local electric and magnetic particles, exchanged by a Z2D self-duality symmetry. We introduce a staggered generalization of the model in which these particles carry global U(1)e and U(1)m charges, respectively, and we propose a continuum QFT description in terms of QED3 with Nf=2 Dirac fermion flavors and a charge-two Higgs field with Yukawa couplings. The conjectured phase diagram harbors a multicritical CFT with (O(2)e×O(2)m)⋊Z2D symmetry, some of which is emergent in the QFT description. We compute the scaling dimensions of some operators using a large-Nf expansion and find agreement with the emergent selection rules. The staggered model admits a deformation to the original Fradkin-Shenker model, which maps to unit-charge monopole operators in Higgs-Yukawa-QED3 that break the U(1)e×U(1)m symmetry. We show explicitly that this deformation reproduces all features of the Fradkin-Shenker phase diagram. Finally, we propose a multicritical duality between Higgs-Yukawa-QED3 and the easy-plane CP1 model (i.e. two-flavor scalar QED3 with a suitable potential), which describes spin-1/2 anti-ferromagnets on a square lattice. This duality implies a first-order line of Néel-VBS transitions ending in a deconfined quantum multicritical point, described by the same O(2)e×O(2)m symmetric CFT that arises in the staggered Fradkin-Shenker model, which separates it from a gapped Z2 spin liquid phase.
In the tensor network representation, a deformed Z2 topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.
We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space R2 in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized "magnetic" flux, with k being the level of the Chern-Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle k2π. Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.