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Register a new userStudy of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
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Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the $((5pmsqrt{5})/2)$-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.
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The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular $mathcal{S}$ matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order.
Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. In this paper, we develop a unified framework based on variational iPEPS for the quantitative study of both topological and conventional phase transitions through entanglement order parameters. To this end, we employ tensor networks with suitable physical and/or entanglement symmetries encoded, and allow for order parameters detecting the behavior of any of those symmetries, both physical and entanglement ones. First, this gives rise to entanglement-based order parameters for topological phases. These topological order parameters allow to quantitatively probe topological phase transitions and to identify their universal behavior. We apply our framework to the study of the Toric Code model in different magnetic fields, which in some cases maps to the (2+1)D Ising model. We identify 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we moreover find an unknown critical exponent beta=0.021. We then apply our framework of entanglement order parameters to conventional phase transitions. We construct a novel type of disorder operator (or disorder parameter), which is non-zero in the disordered phase and measures the response of the wavefunction to a symmetry twist in the entanglement. We numerically evaluate this disorder operator for the (2+1)D transverse field Ising model, where we again recover a critical exponent hitherto unknown in the model, beta=0.024, consistent with the findings for the Toric Code. This shows that entanglement order parameters can provide additional means of characterizing the universal data both at topological and conventional phase transitions, and altogether demonstrates the power of this framework to identify the universal data underlying the transition.
The surfaces of three dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop topological order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $sigma_{xy}=1/2$ and hence break $mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $theta=pi$ magnetoelectric response. Two candidate states have the same 12 particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their $mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24 particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $mathcal T$ symmetric surface state.
Topological phases of matter lie at the heart of physics, connecting elegant mathematical principles to real materials that are believed to shape future electronic and quantum computing technologies. To date, studies in this discipline have almost exclusively been restricted to single-gap band topology because of the Fermi-Dirac filling effect. Here, we theoretically analyze and experimentally confirm a novel class of multi-gap topological phases, which we will refer to as non-Abelian topological semimetals, on kagome geometries. These unprecedented forms of matter depend on the notion of Euler class and frame charges which arise due to non-Abelian charge conversion processes when band nodes of different gaps are braided along each other in momentum space. We identify such exotic phenomena in acoustic metamaterials and uncover a rich topological phase diagram induced by the creation, braiding and recombination of band nodes. Using pump-probe measurements, we verify the non-Abelian charge conversion processes where topological charges of nodes are transferred from one gap to another. Moreover, in such processes, we discover symmetry-enforced intermediate phases featuring triply-degenerate band nodes with unique dispersions that are directly linked to the multi-gap topological invariants. Furthermore, we confirm that edge states can faithfully characterize the multi-gap topological phase diagram. Our study unveils a new regime of topological phases where multi-gap topology and non-Abelian charges of band nodes play a crucial role in understanding semimetals with inter-connected multiple bands.
We propose a unified scheme to identify phase transitions out of the $mathbb{Z}_2$ Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a $mathbb{Z}_2$-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $sigma$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.
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