We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-operator symmetries. The topological phase transition manifests itself by symmetry breaking in the entanglement degrees of freedom of the quantum transfer matrix.
We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-operator symmetries. The topological phase transition manifests itself by symmetry breaking in the entanglement degrees of freedom of the quantum transfer matrix.
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 ((5±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.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of projected entangled pair states, topological order is characterized by an entanglement symmetry of the local tensor that describes the model. This symmetry can take the form of a tensor product of group representations [for quantum double models D(G) of a group G], 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, including those that are not quantum double models. Among other things, these entanglement symmetries allow for the succinct description of ground states and topological excitations (anyons). 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 we use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net model and its Galois conjugate, namely the non-Hermitian double Yang-Lee (DYL) string-net model. 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 we show that these can be mapped to certain restricted solid on solid (RSOS) models, which are equivalent to the $[(5\ifmmode\pm\else\textpm\fi{}\sqrt{5})/2]$-state Potts model, respectively. Moreover, the order parameter for condensation maps to the RSOS order parameter. The known exact solutions of the statistical models subsequently allow us to locate the critical points of the models, and to predict the critical exponents for the order parameters from conformal field theory. 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 behavior of order parameters for condensation and deconfinement.
The non-Abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wave function with two tuning parameters dual with each other, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ϕ^{2}-state Potts models, where ϕ=(sqrt[5]+1)/2 is the golden ratio. By developing the tensor network representation of this wave function on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-Abelian Fibonacci topological phase is enclosed by three distinct nontopological phases and their dual phases of a single ϕ^{2}-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those nontopological phases.
The fundamental concepts of phases of matter and thermal phase transitions constitute the cornerstone of our understanding of the physical universe. The historical development of the phase transition theory from Landau’s spontaneous symmetry breaking paradigm to modern topological phase transition theories represents a major milestone in the evolution of numerous scientific disciplines. From the perspective of emergent philosophy, the interplay of topological excitations leads to enriched physical phenomena. One prominent prototype is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, where unbinding of integer vortices occurs in the absence of spontaneous breaking of continuous U(1) symmetry. Using the state-of-the-art tensor network methods, we express the partition function of the two-dimensional XY-related system in terms of a product of one-dimensional transfer operators. From the singularities of the entanglement entropy of the one-dimensional transfer operator, we accurately determine the complete phase diagram of the partition function. This method provides new insights into the emergent phenomenon driven by topological excitations, and sheds new light on future studies of 2D systems with continuous symmetries.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.
We study the phase diagram of a topological string-net-type lattice model in the presence of geometrically frustrated interactions. These interactions drive several phase transitions that reduce the topological order, leading to a rich phase diagram including both Abelian (Z2) and non-Abelian $(\text{Ising}\ifmmode\times\else\texttimes\fi{}\overline{\text{Ising}})$ topologically ordered phases, as well as phases with broken translational symmetry. Interestingly, one of these phases simultaneously exhibits (Abelian) topological order and long-ranged order due to translational symmetry breaking, with nontrivial interactions between excitations in the topological order and defects in the long-ranged order. We introduce a variety of effective models, valid along certain lines in the phase diagram, which can be used to characterize both topological and symmetry-breaking order in these phases and in many cases allow us to characterize the phase transitions that separate them. We use exact diagonalization and high-order series expansion to study areas of the phase diagram where these models break down and to approximate the location of the phase boundaries.