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2 Models for gapped boundaries and domain walls

Alexei Kitaev, Liang Kong

Published Aug 17, 2016• 431 citations
OpenAlex

Abstract

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

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We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category \calC\calC\calC as in the Levin-Wen model, whereas the boundary is associated with a module category over \calC\calC\calC. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.
A bstract This paper introduces a novel systematic construction of gapped domain walls (GDWs) within the Levin-Wen (LW) model. By gluing two LW models along their open sides in a compatible way, we achieve a complete GDW classification by subsets of bulk input data, which encompass the classifications in terms of bimodule categories. A generalized bimodule structure is introduced to capture domain-wall excitations. Furthermore, we demonstrate that folding along any GDW yields a gapped boundary (GB) described by a Frobenius algebra of the input UFC for the folded model, thus bridging our GDW classification and the GB classification in [1].
In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.
Abstract The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.
Finite depth quantum circuits provide an equivalence relation between gapped phases. Moreover, there can be nontrivial domain walls either within the same gapped phase or between different gapped phases, whose equivalence relations are given by finite depth quantum circuits in one lower dimension. In this paper, we use such unitary equivalence relations to study the fusion of one-dimensional gapped phases. In particular, we use finite depth circuits to fuse two gapped phases, local unitaries to fuse two domain walls, and a combination of both to fuse gapped phases with domain walls. This provides a concrete illustration of some simple aspects of the higher-category structure of gapped defects in a higher-dimensional trivial gapped bulk state.