We investigate domain walls between topologically ordered phases in two spatial dimensions. We present a method which allows for the determination of the superselection sectors of excitations of such walls and which leads to a unified description of the kinematics of a wall and the two phases to either side of it. This incorporates a description of scattering processes at domain walls which can be applied to questions of transport through walls. In addition to the general formalism, we give representative examples including domain walls between the Abelian and non-Abelian topological phases of Kitaev's honeycomb lattice model in a magnetic field, as well as recently proposed domain walls between spin polarized and unpolarized non-Abelian fractional quantum Hall states at different filling fractions.
We investigate domain walls between topologically ordered phases in two spatial dimensions. We present a method which allows for the determination of the superselection sectors of excitations of such walls and which leads to a unified description of the kinematics of a wall and the two phases to either side of it. This incorporates a description of scattering processes at domain walls which can be applied to questions of transport through walls. In addition to the general formalism, we give representative examples including domain walls between the Abelian and non-Abelian topological phases of Kitaev's honeycomb lattice model in a magnetic field, as well as recently proposed domain walls between spin polarized and unpolarized non-Abelian fractional quantum Hall states at different filling fractions.
Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.
Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group G. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as S3) in 2D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.
Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a framework for decomposing stacks of multiple domain walls into superselection sectors — i.e., into fundamental domain wall types that cannot be mixed by any local operators. Such a decomposition can be understood using an alternate framework in the case that the topological order is anomaly-free, in the sense that it can be realized by a commuting projector lattice model. By placing these Witt equivalences in the context of a 3-category of potentially anomalous (2+1)D topological orders, we develop a framework for computing the decomposition of parallel topological domain walls into indecomposable superselection sectors, extending the previous understanding to topological orders with non-trivial anomaly. We characterize the superselection sectors in terms of domain wall particle mobility, which we formalize in terms of tunnelling operators. The mathematical model for the 3-category of topological orders is the 3-category of fusion categories enriched over a fixed unitary modular tensor category.
Transitions between different topologically ordered phases have been studied by artificially creating boundaries between these gapped phases and thus studying their effects relating to condensation and tunneling of particles from one phase to the other. In this work we introduce exactly solvable models which are similar to the quantum double models (QDM) of Kitaev where such domain walls are dynamically generated making them a part of the spectrum. These systems have a local symmetry and may or may not have a global symmetry leading to the possibility of a ground state degeneracy arising from both, global symmetry breaking and a topological degeneracy. The domain wall states now separate the different topologically ordered phases belonging to the different sectors controlled by the global symmetry, when present. They have interesting properties including fusion with the deconfined anyons of the topologically ordered phase, to either create new domain walls, or their annihilation. They can also act like a scatterer permuting anyons of the topologically ordered phases on either side of the domain wall. Thus these domain wall states can be thought of as a synthetic scatterer which can be created in any part of the lattice. We show these effects for the simplest case of the Z2 toric code phase decorated with a global Z2 symmetry and then make remarks about the case when we have the QDM based on two arbitrary groups G1 and G2 in which case we may no longer have a global symmetry.
We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector---referred to as the parton sector---that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground-state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020). Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons K-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the K-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose K matrix is the Cartan matrix of Lie group E8. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.