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.
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.
We consider the string-net model on the honeycomb lattice for Ising anyons in the presence of a string tension. This competing term induces a nontrivial dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown of the topological phase. Using high-order series expansions and exact diagonalizations, we determine the robustness of this doubled Ising phase which is found to be separated from two gapped phases. An effective quantum dimer model emerges in the large tension limit giving rise to two different translation symmetry-broken phases. Consequently, we obtain four transition points, two of which are associated to first-order transitions whereas the two others are found to be continuous and provide examples of recently proposed Bose condensation for anyons.
Abstract A mechanism for constructing one anyon theory from another is anyon condensation, akin to Bose condensation. It is believed that any continuous phase transition between TQFTs must occur by such a condensation transition. We describe the main steps in such a condensation: Identification/Splitting, and Confinement. We give simple examples when the condensing boson is a simple current then describe the more general structure. We explain how the idea of a Chern-Simons coset can be understood under the framework of condensation. Finally we discuss the relationship between condensation and gappable boundaries.
Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in Z(VecGω), i.e. in group-theoretical topological orders for finite groups G with anomaly ω. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory.In particular, topologically ordered phases are topological phases with topological order.Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations.For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons.In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations.These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation).The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases.One important aspect is to understand the braiding statistics in topologically ordered phases.While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear.In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase.In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics.It is a particleloop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase.Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops.Another important aspect is to probe the topological order in topologically ordered phases.One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces.For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons.Also, ground state transformation under twisting the torus encodes the braiding data.In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces.We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics.They in turn distinguish different topological order.Even without topological order, there can be non-trivial topological phases in the presence of global symmetry.These topological phases are called the symmetry protected topological phases.For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological I would like to give my sincere thanks to my advisor Shinsei Ryu.Throughout my PhD study, he taught me a lot about condensed matter physics, particularly on the forefront subjects in topological phases.Under his supervision, I worked on different abstract yet inspiring topics on topological phases.His guide is always clear with mathematical interpretation and physical insight.His advice not only guided me though the research, but also shaped my views and thinking.From my advisor, I learnt the way to convey difficult ideas
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory. In particular, topologically ordered phases are topological phases with topological order. Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations. For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons. In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations. These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation). The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases. One important aspect is to understand the braiding statistics in topologically ordered phases. While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear. In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase. In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics. It is a particle-loop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase. Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops. Another important aspect is to probe the topological order in topologically ordered phases. One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces. For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons. Also, ground state transformation under twisting the torus encodes the braiding data. In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces. We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics. They in turn distinguish different topological order. Even without topological order, there can be non-trivial topological phases in the presence of global symmetry. These topological phases are called the symmetry protected topological phases. For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.
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.