Gapless Coulomb State Emerging from a Self-Dual Topological Tensor-Network State

Abstract

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.Introduction.-The toric code model proposed by Kitaev[1] is a prototypical model realizing the Z 2 intrinsic topological phase of matter with anyonic excitations. It is interesting and fundamentally important to consider the possible topological phase transitions out of the toric code phase, because such phase transitions are beyond the conventional Ginzburg-Landau paradigm for the symmetry breaking phases. From the perspective of lattice gauge theory, it has been known that there exists the Higgs/confinement transition, where the electric charge is condensed/confined accompanied by the confinement/condensation of magnetic flux due to electricmagnetic duality [2][3][4][5][6][7]. However, there is a long-standing puzzle: what is the nature of the phase transition along the self-dual line and how the Higgs and confinement transition lines merge into the self-dual phase transition point [6,8]. Should there be a tricritical point, it would go beyond the anyon condensation scenario[9], because the electric charge and magnetic flux are not allowed to simultaneously condense.In this Letter, we shall resolve this puzzle and provide new insight into the nature of this topological phase transition. Instead of solving a Hamiltonian with tuning parameters, we propose a deformed topological wave-function interpolating from the nontrivial to trivial phases in the tensor network representation [10][11][12], which provides a clearer scope into the essential physics of abelian anyonic excitations [13][14][15][16][17]. In this scheme, the usual pure Higgs/confinement transition of the toric code [14,18,19] has a special path, where the deformed wave-function can be exactly mapped to a twodimensional (2D) classical Ising model. The topological phase transition is associated with the 2D Ising phase transition, drawing the striking topological-symmetrybreaking correspondence [20].Further deformation of the toric code wave functions can span a generalized phase diagram [21][22][23], where the perturbed Higgs and confinement transitions were gener-ically obtained by the symmetry breaking pattern and long-range-...

Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions

“…1c. These auxiliary Ising spins contain the crucial information about anyonic excitations 22,23 . Importantly, when the domain-walls of the Ising paramagnet are assigned orientations according to the rule displayed in Fig.…”

“…which can be regarded as a generic DS wavefunction in an expanded parameter space. Actually, the similar deformation has been used to express the generic TC wavefunction [23][24][25] and Fibonacci quantum-net wavefunction 26 . For convenience, we define h x h cos θ and h z h sin θ, where h expresses the loop tension and θ is the spin angle.…”

“…It has been proved that 23 the partial ordered phase, ferromagnetic phase, and paramagnetic phase of the Ashkin-Teller model correspond to the TC phase, Higgs phase, and confining phase, respectively. So a global phase diagram can be derived as displayed The phase transition lines JH and JKF belong to the two-dimensional Ising universality class, and the line IJ is characterized by the Kosterlitz-Thouless transition.…”

Variational methods for characterizing matrix product operator symmetries

“…In order to drive the iPEPS away from a fixed point and introduce a finite correlation length we apply the local filtering [69][70][71] to the fixed point of the Fibonacci stringnet model. The modification has the following form:…”

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

“…Here, the topological order is accompanied by the presence of certain group or Matrix Product Operator (MPO) symmetries in the entanglement degrees of freedom of the tensor, which can be used to parametrize the ground space manifold and anyonic excitations alike [7][8][9]. The description of topologically ordered systems as PEPS based on entanglement symmetries suggests a natural way to construct and study topological phase transitions within PEPS, by applying deformations to the physical degrees of freedom which drive the system to a different phase (such as a trivial product state) [10][11][12][13][14][15][16][17]. In this language, the entanglement symmetry in the tensor is preserved throughout the path, but at some point, it no longer manifests itself in topological order.…”