In the tensor network representation, a deformed $Z_{2}$ 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.
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.
In the tensor network representation, a deformed Z_{2} topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the Ashkin-Teller 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 the confining phase.
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.
Topological phase transition of deformed Z3 toric code
We investigate topological phase transitions in a family of deformed Z3 toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the Q=3 Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT3) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric (e) anyons are confined or condensed. These phases are separated by critical structures with central charges c=4/5, 8/5, and isolated c=1 antiferromagnetic (AFM) endpoints. A normalized finite-distance e-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double D(Z3) imply a topological entanglement entropy γ=log3 throughout the gapped toric-code phase. Relative to the Z2 case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent U(1) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.
Due to the absence of local order parameters, it is a challenging task to characterize the quantum topological phase transitions between topologically ordered phases in two dimensions. In this paper, we construct a topologically ordered tensor network wave function with one parameter $\ensuremath{\lambda}$, describing both the toric code state $(\ensuremath{\lambda}=1)$ and double semion state $(\ensuremath{\lambda}=\ensuremath{-}1)$. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\ensuremath{\lambda}$, and three different quantum critical points (QCPs) at $\ensuremath{\lambda}=0,\ifmmode\pm\else\textpm\fi{}1.73$ are identified. The first one separates the toric code phase and double semion phase, while latter two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry-breaking phase, respectively. When mapping the quantum tensor network wave function to the exactly solved statistical model, the norm of the wave function is identified as the partition function of the classical eight-vertex model, and both QCPs at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1.73$ correspond to the eight-vertex model at the critical point $\ensuremath{\lambda}=\sqrt{3}$, while the QCP at $\ensuremath{\lambda}=0$ corresponds to the critical six-vertex model. Actually such a quantum-classical mapping cannot yield the complete low-energy excitations at these three QCPs. We further demonstrate that the full eigenvalue spectra of the transfer operators without/with the flux insertions can give rise to the complete quantum criticalities, which are described by the (2+0)-dimensional free boson conformal field theories (CFTs) compactified on a circle with the radius R=6 at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}$ and R=8/3 at $\ensuremath{\lambda}=0$. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at $\ensuremath{\lambda}=0$, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.
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.