Anyon Condensation Research Articles

Two-dimensional topological order and operator algebras

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

Vestigial anyon condensation in kagome quantum spin liquids

We construct a lattice model of topological order (kagome quantum spin liquids) and solve it with unbiased quantum Monte Carlo simulations. A three-stage anyon condensation with two transitions from a $\mathbb Z_2\boxtimes\mathbb Z_2$ topological order to a $\mathbb Z_2$ topological order and eventually to a trivial symmetric phase is revealed. These results provide concrete examples of phase transitions between topological orders in quantum magnets. The designed quantum spin liquid model and its numerical solution offer a playground for further investigations on vestigial anyon condensation.

Superuniversality from disorder at two-dimensional topological phase transitions

We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (`anyons') identified with `electric charge' excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is `superuniversal' across a wide range of such condensation transitions, and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.

Particle-hole Pfaffian order in a translationally and rotationally invariant system

The PH-Pfaffian topological order has been proposed as a candidate order for the $\nu=5/2$ quantum Hall effect. The PH-Pfaffian liquid is known to be the ground state in several coupled wire and coupled stripe constructions. No translationally and rotationally invariant models with the PH-Pfaffian ground state have been identified so far. By employing anyon condensation on top of a topological order, allowed in an isotropic system, we argue that the PH-Pfaffian order is possible in the presence of rotational and translational symmetries.

Entanglement entropy of generalized Moore-Read fractional quantum Hall state interfaces

Topologically ordered phases of matter can be characterized by the presence of a universal, constant contribution to the entanglement entropy known as the topological entanglement entropy (TEE). The TEE can been calculated for Abelian phases via a "cut-and-glue" approach by treating the entanglement cut as a physical cut, coupling the resulting gapless edges with explicit tunneling terms, and computing the entanglement between the two edges. We provide a first step towards extending this methodology to non-Abelian topological phases, focusing on the generalized Moore-Read (MR) fractional quantum Hall states at filling fractions $\nu=1/n$. We consider interfaces between different MR states, write down explicit gapping interactions, which we motivate using an anyon condensation picture, and compute the entanglement entropy for an entanglement cut lying along the interface. Our work provides new insight towards understanding the connections between anyon condensation, gapped interfaces of non-Abelian phases, and TEE.

Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries

We generalize the electric-magnetic (EM) duality in the quantum double (QD) models to the extended QD models of topological orders with gapped boundaries. We also map the extended QD models to the extended Levin-Wen (LW) models with gapped boundaries. To this end, we Fourier-transform and rewrite the extended QD model on a trivalent lattice with a boundary, where the bulk gauge group is a finite group G. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups K ⊆ G supplying the boundary degrees of freedom. We find that after the transformation, the boundary conditions are then characterized by the Frobenius algebras AG,K in RepG. An AG,K is the dual space of the quotient of the group algebra of G over that of K , and RepG is the category of the representations of G. The EM duality on the boundary is revealed by mapping the K ’s to AG,K ’s. We also show that our transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the latter. Moreover, our transformed extended QD model elucidates the phenomenon of anyon splitting in anyon condensation.

Unhinging the Surfaces of Higher-Order Topological Insulators and Superconductors.

We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting C_{2n}T symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time-reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a C_{2n}T preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between C_{2n}T-conjugate domains. However, using a combination of K-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence "anomalous." Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.

Anisotropic layer construction of anisotropic fracton models

We propose a coupled-layer construction of a class of fracton topological orders in three spatial dimensions, which is characterized by spatially anisotropic mobility of quasiparticle excitations constrained in subdimensional manifolds. The simplest model is obtained by stacking and coupling layers of the two-dimensional toric codes on the square lattice and can be exactly solved in the strong-coupling limit. The resulting fracton excitations are understood as a consequence of anyon pair condensation induced by the coupling between layers. We also present generalizations of the construction for layers of the Kitaev-honeycomb models, the $Z_N$ toric codes, and the toric codes and the doubled semion models on the honeycomb lattice.

From coupled wires to coupled layers: Model with three-dimensional fractional excitations

We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the (1+1)-dimensional conformal field theory (CFT) associated with a current algebra $\mathfrak{g}$. The wires are coupled with each other to form a 3D network through the current-current interactions of $\mathfrak{g}_1$ and $\mathfrak{g}_2$ CFTs that are related to the $\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g} \supset \mathfrak{g}_1 \times \mathfrak{g}_2$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create point-like or loop-like deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU(2)_1 \times SU(2)_1 \supset U(1)_4 \times U(1)_4$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D $Z_2$ gauge theory with a fermionic $Z_2$ charge.

Cage-Net Fracton Models

We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux-strings" built out of point-like excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton ordered (equivalently, flux-string condensed) phase, such as its ground state wave function and spectrum of excitations. Through the examples of doubled Ising and SU(2)$_k$ cage-net models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension. In the doubled Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states amongst two-dimensional non-Abelian anyons and Abelian particles. We further show that the ground state wave function of such phases can be understood as a fluctuating network of extended objects -- cages -- and strings, which we dub a cage-net condensate. Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.

Spontaneous symmetry breaking from anyon condensation

In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a G-crossed braided extension mathcal{C}subseteq {mathcal{C}}_G^{times } , we show that physical considerations require that a connected étale algebra A ∈ mathcal{C} admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action G → EqBr( mathcal{C} ) such that g(A) ≅ A for all g ∈ G, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of {mathcal{C}}_A^{mathrm{loc}} , and gauging this symmetry commutes with anyon condensation.

Building symmetry-enriched topological phases from a bipartite lattice construction and anyon condensation

We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two $\mathbb{Z}_2$ spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton based approaches. We demonstrate the construction for $i$) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, $ii$) a nonsymmorphic lattice, and $iii$) lattices with magnetic translation symmetry. At last, we discuss constraints on non-chiral topological orders in a bosonic system under magnetic field.

Condensation of non-Abelian anyons in a one-dimensional fermion model

The color excitations of interacting fermions carrying an color and U(Nf) flavor index in one spatial dimension are studied in the framework of a perturbed Wess–Zumino–Novikov–Witten model. Using Bethe ansatz methods the low energy quasi-particles are found to be massive solitons forming quark and antiquark multiplets. In addition to the color index the solitons carry an internal degree of freedom with non-integer quantum dimension. These zero modes are identified as non-Abelian anyons satisfying fusion rules. Controlling the soliton density by external fields allows to drive the condensation of these anyons into various collective states. The latter are described by parafermionic cosets related to the symmetry of the system. Based on the numerical solution of the thermodynamic Bethe ansatz equations we propose a low temperature phase diagram for this model.

Tensor network state approach to quantum topological phase transitions and their criticalities of Z2 topologically ordered states

We construct a general wave function with the topological order by introducing the $\mathbb{Z}_{2}$ gauge degrees of freedom, characterizing both the toric code state and double semion state. 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 $\lambda $ and identify three different quantum critical points (QCPs) at $\lambda =0$, $\pm 1.73$. The first one separates the toric code phase and double semion phase, while later two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry breaking phase, respectively. When mapping to the exactly solved statistical models, the norm of the tensor network wave function is transformed into the partition function of the eight-vertex model. Actually such a quantum-classical mapping can not reveal the rich structures of low-energy excitations at these three QCPs. So we further demonstrate that the full eigenvalue spectra of the transfer operators with/without the flux insertions can describe the complete quantum criticalities, which are characterized by the two-dimensional compactified free boson conformal field theories (CFTs) with the compactified radii $R=\sqrt{6}$ for the QCPs at $\lambda =\pm\sqrt{3}$ and $R=\sqrt{8/3}$ for the QCP at $\lambda =0$. For the QCP at $\lambda =0$, there are no anyon condensation, and the emerged matrix product operator symmetries result in a rich structure of the low-energy excitations, distinct from those of both toric code and double semion phases. Finally, we discuss the possible relation between our conformal quantum criticalities and the general (2+1) spatial-time dimensional CFTs for quantum topological phase transitions.

Study of anyon condensation and topological phase transitions from a Z4 topological phase using the projected entangled pair states approach

We use Projected Entangled Pair States (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons, and serve as a substitute for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a $\mathbb Z_4$ symmetry, which contains $\mathbb Z_4$ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.

Condensation-driven phase transitions in perturbed string nets

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.

Symmetry reduction induced by anyon condensation: A tensor network approach

Topological ordered phases are related to changes in the properties of their quasi-particle excitations (anyons). We study these relations in the framework of projected entanglement pair states (\textsf{PEPS}) and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well, and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states (\textsf{MPS}). The case of $\mathbb{Z}_2$ on-site symmetry is studied in detail.

Anyon Condensation and Its Applications

Bose condensation is central to our understanding of quantum phases of matter. Here, we review Bose condensation in topologically ordered phases (also called topological symmetry breaking), where the condensing bosons have nontrivial mutual statistics with other quasiparticles in the system. We give a nontechnical overview of the relationship between the phases before and after condensation, drawing parallels with more familiar symmetry-breaking transitions. We then review two important applications of this phenomenon. First, we describe the equivalence between such condensation transitions and pairs of phases with gappable boundaries, as well as examples where multiple types of gapped boundary between the same two phases exist. Second, we discuss how such transitions can lead to global symmetries that exchange or permute anyon types. Finally, we discuss the nature of the critical point, which can be mapped to a conventional phase transition in some—but not all—cases.

Entanglement phases as holographic duals of anyon condensates

Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the Z4 double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.