The Quantum Double Model with Boundary: Condensations and Symmetries

Beigi, Salman; Whalen, Daniel

doi:10.1007/s00220-011-1294-x

Cited by 166 publications

(290 citation statements)

References 16 publications

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“…This might result in the simple permuting-type defect lines introducing entropic protection to Abelian systems. Our analysis only applies to Abelian quantum doubles, i.e., qudit stabilizer codes, therefore, entropic protection of quantum doubles based on non-Abelian groups (or of models that are not quantum doubles of any group) is not ruled out, especially since one can think of a variety of defect lines which can arise in such models [27]. One can also consider constructions where lower-dimensional topological systems are coupled to an ancillary system, and this coupling modifies the dynamics of the original model [5,28].…”

Section: Discussionmentioning

confidence: 99%

Necessity of an energy barrier for self-correction of Abelian quantum doubles

2016

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We rigorously establish an Arrhenius law for the mixing time of quantum doubles based on any Abelian group Z d . We have made the concept of the energy barrier therein mathematically well defined; it is related to the minimum energy cost the environment has to provide to the system in order to produce a generalized Pauli error, maximized for any generalized Pauli errors, not only logical operators. We evaluate this generalized energy barrier in Abelian quantum double models and find it to be a constant independent of system size. Thus, we rule out the possibility of entropic protection for this broad group of models.

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Section: Discussionmentioning

confidence: 99%

Necessity of an energy barrier for self-correction of Abelian quantum doubles

2016

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“…We will be most interested in studying these defects in explicit lattice models [6][7][8][9][10], and in particular the model introduced by Bombin [6]. Essentially the same defects have also been studied in Chern-Simons theories [11][12][13].…”

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confidence: 99%

Topological Entanglement Entropy with a Twist

et al. 2013

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Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that are analogous to Ising anyons. We strengthen this analogy by using the topological entanglement entropy as a diagnostic tool to identify properties of both defects and excitations in the toric code. Specifically, we show, through explicit calculation, that the toric code model including twists and dyon excitations has the same quantum dimensions, the same total quantum dimension, and the same fusion rules as an Ising anyon model.A fascinating class of many-body quantum systems are those that exhibit topological order [1]. Such systems are characterized by a gapped ground state manifold, with a degeneracy that depends on the boundary conditions, and have anyonic quasi-particle excitations. The degeneracy is robust to local perturbations, and therefore such systems are promising candidates for storing and manipulating quantum information [2][3][4][5].The structure of topologically-ordered systems can be further enriched by the use of domain walls or defects, across which quasi-particles transform nontrivially. We will be most interested in studying these defects in explicit lattice models [6][7][8][9][10], and in particular the model introduced by Bombin [6]. Essentially the same defects have also been studied in Chern-Simons theories [11][12][13]. In both of these settings, it has been shown that such defects can be viewed as having anyonic-like properties that are not associated with the underlying model [6,9,[11][12][13]. More specifically, in a particular two-dimensional topologically ordered spin lattice model known as the toric code, such a lattice dislocation leads to interesting behaviour: the points where dislocations terminate, known as twists, interact with the anyons of the toric code to reproduce properties, such as fusion rules, of nonabelian (specifically, Ising) anyons [6]. This is particularly surprising, as all excitations of the toric code are abelian anyons.In this paper, we further interrogate the analogy between twists and Ising anyons by using topological entanglement entropy (TEE) [14,15] as a diagnostic. Specifically, for a toric code model containing twists, we use the TEE to determine: (i) the total quantum dimension of the lattice with twists; (ii) the quantum dimensions of the objects (quasi-particles and defects) on the lattice, and (iii) the quantum dimensions of all of their fusion products, allowing us to reconstruct the fusion rules for twists and excitations. Our results coincide precisely with those of the Ising anyon model, lending further support to the analogy between the latter and the toric code with twists. We note that TEE is a particularly useful quantity in the context of the toric code, as both the parent Hamiltonian and the modified one with twists are described within the stabilizer formalism and so allow the TEE to be calculated e...

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“…Third, in the case with finite groups, the Levin-Wen model is dual to the Kitaev quantum double model. Shor et al have shown that the boundary degrees of freedom in the Kitaev model defined by a finite group G live in a subgroup of G [37], which in the dual Levin-Wen model corresponds to a Frobenius algebra in the UFC of the representations of G.…”

Section: A Gapped Energy Spectrum For Each Boundary Hamiltonian Togetmentioning

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%

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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The Quantum Double Model with Boundary: Condensations and Symmetries

Cited by 166 publications

References 16 publications

Necessity of an energy barrier for self-correction of Abelian quantum doubles

Necessity of an energy barrier for self-correction of Abelian quantum doubles

Topological Entanglement Entropy with a Twist

Boundary Hamiltonian theory for gapped topological phases on an open surface

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