We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension theory symmetry-breaking phase which applies to with topological excitations described quantum groups or modular tensor categories. This enables us deal whose quasiparticles have noninteger and obey braid statistics. Many examples such can be constructed from two-dimensional rational conformal field theories,...

We generalize the fractional quantum Hall hierarchy picture to apply arbitrary, possibly non-Abelian, states. Applying this $\ensuremath{\nu}=5/2$ Moore-Read state, we construct explicit trial wave functions describe effect in second Landau level. The resulting of states, which reproduces filling fractions all observed conductance plateaus level, is characterized by electron pairing ground state and an excitation spectrum that includes non-Abelian anyons Ising type. propose as a unifying...

We investigate domain walls between topologically ordered phases in two spatial dimensions. present a method which allows for the determination of superselection sectors excitations such and leads to unified description kinematics wall either side it. This incorporates scattering processes at can be applied questions transport through walls. In addition general formalism, we give representative examples including Abelian non-Abelian topological Kitaev's honeycomb lattice model magnetic...

We consider the tunneling current through a double point-contact Fabry-P\'erot interferometer such as used in recent experimental studies of fractional quantum Hall plateau at filling fraction $\ensuremath{\nu}=5/2$. compare predictions several different models state electrons this plateau: Moore-Read, anti-Pfaffian, $\text{SU}{(2)}_{2}$ NAF, $K=8$ strong pairing, and (3,3,1) states. All these predict existence charge $e/4$ quasiparticles, but first three are non-Abelian while last two...

We describe a family of phase transitions connecting phases differing non-trivial topological order by explicitly constructing Hamiltonians the Levin-Wen[PRB 71, 045110] type which can be tuned between two solvable points, each realizes different topologically ordered phase. show that low-energy degrees freedom near transition mapped onto those Potts model, and we discuss stability resulting diagram to small perturbations about model. further explain how excitations in condensed are formed...

We examine interferometric experiments in systems that exhibit non-Abelian braiding statistics, expressing outcomes terms of the modular S-matrix. In particular, this result applies to fractional quantum Hall interferometry, and we give a detailed treatment Read-Rezayi states, providing explicit predictions for recently observed nu = 12/5 plateau.

We examine how best to design qubits for use in topological quantum computation. These are Hilbert spaces associated with small groups of anyons. Op- erations performed on these by exchanging the One might argue that, order have as many simple single qubit operations possible, number anyons per group should be maximized. However, we show that there is a maximal particles qubit, namely 4, and more generally qudits dimension d. also look at possibility having which one can perform two-qubit...

We devise a way to calculate the dimensions of symmetry sectors appearing in particle entanglement spectrum (PES) and real space (RSES) multiparticle systems from their wave functions. first note that these ranks spectra equal spaces functions with number particles fixed. This also yields equality multiplicities PES RSES. Our technique allows numerical calculations for much larger than were previously feasible. For somewhat smaller systems, we can find approximate energies as well...

We study transitions between phases of matter with topological order. By studying these in exactly solvable lattice models we show how universality classes may be identified and critical properties described. As a familiar example to elucidate our results concretely, describe detail transition fully gapped achiral 2D $p$-wave superconductor ($p+ip$ for pseudospin up/$p-ip$ down) an $s$-wave which the transverse field Ising class.

We study the nonequilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one their defining features: ground-state degeneracies and associated sectors. present notion ``topological blocking,'' experienced by due to mismatch between two phases, we argue that dynamic evolution quench depends strongly sector being probed. demonstrate this interplay topology models stemming from extensively studied transverse Ising chain Kitaev honeycomb model....

The braiding of the worldlines particles restricted to move on a network (graph) is governed by graph braid group, which can be strikingly different from standard group known two-dimensional physics. It has been recently shown that imposing compatibility with anyon fusion for anyons exchanging at single wire junction leads new types models exchange operators stemming solutions certain generalised hexagon equations. In this work, we establish these graph-braided general networks. We show...

Many two-dimensional physical systems have symmetries which are mathematically described by quantum groups (quasitriangular Hopf algebras). In this Letter we introduce the concept of a spontaneously broken symmetry and show that it provides an effective tool for analyzing wide variety phases exhibiting many distinct confinement phenomena.

We present a solution of Kitaev's spin model on the honeycomb lattice and related topologically ordered models. employ Jordan-Wigner type fermionization find that Hamiltonian takes BCS form, allowing system to be solved by Bogoliubov transformation. Our does not non-physical auxiliary degrees freedom eigenstates we obtain are completely explicit in terms variables. The ground-state is obtained as condensate fermion pairs over vacuum state which corresponds toric code with same vorticity....

We provide numerical evidence that a p_{x}-i p_{y} paired Bonderson--Slingerland (BS) non-Abelian hierarchy state is strong candidate for the observed nu=12/5 quantum Hall plateau. confirm existence of gapped incompressible nu = 12/5 with shift S=2 on sphere, matching BS state. The exact ground Coulomb interaction at shown to have large overlap trial wave function. Larger overlaps are obtained BS-type functions hierarchical descendants general weakly-paired states nu=5/2. perform finite size...

Resonance effects lead to surprising and beautiful phenomena in many areas of physics. Here, the authors identify resonance points as key understanding breakdown revival strong zero modes parafermionic clock models. Through analytic numerical methods, they have been able determine behavior at away from points, proving statements about asymptotic these systems. In particular, show that there exists special parameter space where can exist and, such, topological degeneracy is preserved all energies.

The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the Abelian topological degeneracy. We derive perturbative low-energy effective Hamiltonian that is valid all orders expansion and for possible toroidal configurations. Using this form we demonstrate at what order system's degeneracy lifted by finite size effects note thermodynamic limit it robust orders. Further, themselves correspond creation, propagation, annihilation fermions....

We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both bosons fermions, study these numerically. Our construction employs a picture bosonic state as symmetrized double layer composite fermion state. obtain number independent angular momentum multiplets states systems up to 20 electrons. find that counting quasielectrons at large on sphere matches expected from conformal field theory (CFT) describes state's...

We study the entanglement spectra of many-particle systems in states which are closely related to products Slater determinants or permanents, combinations two. Such notably include Laughlin and Jain composite fermion describe most observed conductance plateaus fractional quantum Hall effect. identify a set wave functions (EWFs), for subsets particles, completely such product functions, both real space particle space. A subset EWFs can be recognized as states. These provide an exact...

We have studied topology and dynamics of quantum vortices in spin-2 Bose-Einstein condensates. By computationally modeling controllable braiding fusion these vortices, we demonstrated that certain such spinor condensates behave as non-Abelian anyons. identify anyons fluxon, chargeon, dyon quasiparticles. The pertinent anyon models are defined by the double underlying discrete symmetry group condensate ground state order parameter.

We investigate dynamical evolution of a topological memory that consists two $p$-wave superconducting wires separated by nontopological junction, focusing on the primary errors (i.e., qubit loss) and secondary (bit phase flip) arise due to nonadiabaticity. On question loss we examine system's response both periodic boundary driving deliberate shuttling Majorana bound states. In former scenario, show how frequency-dependent rate is strongly correlated with local density states at edge wire,...

We calculate (q-deformed) Clebsch-Gordan and 6j-coefficients for rank two quantum groups. explain in detail how such calculations are done, which should allow the reader to perform similar other cases. Moreover, we tabulate q-Clebsch-Gordan explicitly, as well some topological data associated with theories corresponding rank-two Finally, collect useful properties of fusion rules particular conformal field theories.

Exactly solvable lattice models for spins and non-interacting fermions provide fascinating examples of topological phases, some them exhibiting the localized Majorana that feature in proposals quantum computing. The Chern invariant ν is an important characterization such phases. Here we look at square–octagon variant Kitaev's honeycomb model. It maps to spinful paired enjoys a rich phase diagram featuring distinct Abelian non-Abelian phases with = 0,±1,±2,±3 ±4. ±1 ±3 all support modes are...