Models for Gapped Boundaries and Domain Walls

Kitaev, Alexei; Kong, Liang

doi:10.1007/s00220-012-1500-5

Cited by 386 publications

(647 citation statements)

References 41 publications

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“…This formulation is closely related to Kitaev and Kong's work [17] that formulates boundary theories using module categories over C. In this section we will discuss the relation between our approach and the Kitaev-Kong (KK) formulation. In our approach, we take boundary degrees of freedom from the labels of the input UFC -the same degrees of freedom as in the bulk, and we start with local boundary Hamiltonians.…”

Section: Relation To the Kitaev-kong Formulationmentioning

confidence: 99%

“…It is worth noting that there has been a few studies of the boundary Hamiltonians in the Kitaev model [36][37][38][39], as well as a study of the Levin-Wen model [17] with boundary in the language of module categories. While our approach is systematic and easier to access by the condensed matter community, we shall discuss in section 10 the relation between our approach and the one taken by Kitaev and Kong [17].…”

Section: Jhep01(2018)134mentioning

confidence: 99%

“…While our approach is systematic and easier to access by the condensed matter community, we shall discuss in section 10 the relation between our approach and the one taken by Kitaev and Kong [17]. The appendices collect certain details and proofs.…”

Section: Jhep01(2018)134mentioning

confidence: 99%

“…Yet, any boundary massless modes that often appear must be gapped to have a well defined GSD. The gapping conditions of Abelian topological orders have recently been understood in terms of the concept of Lagrangian subsets [17][18][19][20][21][22][23][24], and subsequently the GSD of these Abelian phases on open surfaces with multiple boundaries were computed [23,25], based on the idea of anyon transport across boundaries. Experiments detecting and utilizing the topological degeneracy with gapped boundaries were proposed in [26,27].…”

Section: Jhep01(2018)134 1 Introduction and Summarymentioning

confidence: 99%

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Boundary Hamiltonian theory for gapped topological phases on an open surface

Luo

Pankovich

et al. 2018

J. High Energ. Phys.

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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.

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Section: Relation To the Kitaev-kong Formulationmentioning

confidence: 99%

Section: Jhep01(2018)134mentioning

confidence: 99%

Section: Jhep01(2018)134mentioning

confidence: 99%

Section: Jhep01(2018)134 1 Introduction and Summarymentioning

confidence: 99%

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Boundary Hamiltonian theory for gapped topological phases on an open surface

Luo

Pankovich

et al. 2018

J. High Energ. Phys.

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“…More precisely, these are integrated fluxes associated to a closed path. Note that in the almost gauge invariant phase space, discussed in the previous section, we only had fluxes associated to open paths, 33 provided we assume that the graph does not include loops (i.e. links with the same source and target node).…”

Section: Coarse-graining In the Fusion Basismentioning

confidence: 99%

Fusion basis for lattice gauge theory and loop quantum gravity

Delcamp

Dittrich

Riello

2017

J. High Energ. Phys.

133

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We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in (2 + 1) dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gauß constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and (2 + 1) gravity coupled to point particles. In a follow-up work, we have exploited this notion to provide a new definition of entanglement entropy for these theories.

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