Twisted quantum double model of topological phases in two dimensions

Hu, Yuting; Wan, Yidun; Wu, Yong‐Shi

doi:10.1103/physrevb.87.125114

Cited by 147 publications

(221 citation statements)

References 46 publications

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“…For readers who are familiar with SPT phases, it may be clear that the characterization of a wave function in the excitation basis in the fluxless subspace H no-flux is essentially equivalent to "ungauging" the on-site symmetries, opposite to the procedure of gauging the on-site symmetries of the SPT wave functions [14,30]. In this picture of ungauging, the two-dimensional color code with a looplike excitation serves as a path integral formulation of a one-dimensional SPT wave function.…”

Section: A Looplike Excitation From Membrane Operatormentioning

confidence: 99%

Topological color code and symmetry-protected topological phases

Yoshida

2015

Phys. Rev. B

133

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We study (d − 1)-dimensional excitations in the d-dimensional color code that are created by transversal application of the R d phase operators on connected subregions of qubits. We find that such excitations are the superpositions of electric charges and can be characterized by the fixed-point wave functions of (d − 1)-dimensional bosonic symmetry-protected topological (SPT) phases with (Z 2 ) ⊗d symmetry. While these SPT excitations are localized on (d − 1)-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the nontriviality of emerging SPT wave functions. Moreover, these SPT excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal R 3 operator, exchanges a magnetic flux and a composite of a magnetic flux and the looplike SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and the looplike SPT excitations exhibit nontrivial three-loop braiding statistics in three dimensions as a result of the fact that the R 3 phase operator belongs to the third level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.

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Section: A Looplike Excitation From Membrane Operatormentioning

confidence: 99%

Topological color code and symmetry-protected topological phases

Yoshida

2015

Phys. Rev. B

133

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“…We now present here a fascinating example-the Z 3 2 twisted quantum double (TQD) [28][29][30] -that bears more than one set of nontrivial gapped boundary conditions. As a twisted version of the G = Z 3 2 Kitaev model, this model contains 22 distinct anyons.…”

Section: Z 3 2 Twisted Quantum Doublementioning

confidence: 99%

Ground-State Degeneracy of Topological Phases on Open Surfaces

Hung

Wan

2015

Phys. Rev. Lett.

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We relate the ground state degeneracy (GSD) of a non-Abelian topological phase on a surface with boundaries to the anyon condensates that break the topological phase to a trivial phase. Specifically, we propose that gapped boundary conditions of the surface are in one-to-one correspondence to the sets of condensates, each being able to completely break the phase, and we substantiate this by examples. The GSD resulting from a particular boundary condition coincides with the number of confined topological sectors due to the corresponding condensation. These lead to a generalization of the Laughlin-Tao-Wu (LTW) charge-pumping argument for Abelian fractional quantum Hall states (FQHS) to encompass non-Abelian topological phases, in the sense that an anyon loop of a confined anyon winding a non-trivial cycle can pump a condensed anyon from one boundary to another. Such generalized pumping may find applications in quantum control of anyons, eventually realizing topological quantum computation.

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“…Therefore, Hamiltonian models of topological orders such as the Levin-Wen model [32], the Kitaev Model [7], and the twisted quantum double model [33] as a generalization of the Kitaev model are not complete dynamical models on open surfaces because they do not include boundary terms.…”

Section: Jhep01(2018)134 1 Introduction and Summarymentioning

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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Twisted quantum double model of topological phases in two dimensions

Cited by 147 publications

References 46 publications

Topological color code and symmetry-protected topological phases

Topological color code and symmetry-protected topological phases

Ground-State Degeneracy of Topological Phases on Open Surfaces

Boundary Hamiltonian theory for gapped topological phases on an open surface

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