Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

Williamson, Dominic J.; Bultinck, Nick; Verstraete, Frank

doi:10.48550/arxiv.1711.07982

Cited by 42 publications

(80 citation statements)

References 29 publications

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“…Mathematically, this is guaranteed by the fact that the fusion categories D and C are Morita equivalent; the sectors are given by the monoidal centers of these fusion categories, which are equivalent for Morita equivalent fusion categories [22]. At the level of the MPO symmetries, the monoidal center can be constructed from the tube algebra [14,66], the central idempotents of which correspond to the different sectors in the model. The dimension of these central idempotents is in general different for the two models, which is reflected in the difference in Hilbert space dimension.…”

Section: Iid Bond Algebras and Dualitymentioning

confidence: 99%

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Lootens¹,

Delcamp²,

Ortíz³

et al. 2021

Preprint

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We present a systematic approach for generating duality transformations in quantum lattice models. Within our formalism, dualities are completely characterized by equivalent but distinct realizations of a given (possibly non-abelian and non-invertible) symmetry. These different realizations are encoded into fusion categories, and dualities are methodically generated by considering all Morita equivalent categories. The full set of symmetric operators can then be constructed from the categorical data. We construct explicit intertwiners, in the form of matrix product operators, that convert local symmetric operators of one realization into local symmetric operators of its dual. Concurrently, it maps local operators that transform non-trivially into non-local ones. This guarantees that the structure constants of the algebra of all symmetric operators are equal in both dual realizations. Families of dual Hamiltonians, possibly with long range interactions, are then designed by taking linear combinations of the corresponding symmetric operators. We illustrate this approach by establishing matrix product operator intertwiners for well-known dualities such as Kramers-Wannier and Jordan-Wigner, consider theories with two copies of the Ising category symmetry, and present an example with quantum group symmetries. Finally, we comment on generalizations to higher dimensions of this categorical approach to dualities.

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Section: Iid Bond Algebras and Dualitymentioning

confidence: 99%

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Lootens¹,

Delcamp²,

Ortíz³

et al. 2021

Preprint

Self Cite

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“…where |P i is anyon basis. Explicit wave-functions for unitary models based on a tensor network have been constructed explicitly [18][19][20][21][22][23][24]. We would like to adapt these methods to construct left/right anyon basis in the non-Hermitian model.…”

Section: Modular S T Matrix For Non-hermitian Galois Conjugate String...mentioning

confidence: 99%

Galois conjugates of String-net Model

Chen,

Lao,

et al. 2021

Preprint

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We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate bi-orthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the bi-orthogonal basis the topological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates. a

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“…The multiplication of the basis elements e i := A abcd defines some algebra, from which we find both central and simple idempotents as desrcibed in Appendix B. The algebra of A abcd can be used to calculate the action of the Dehn twist on a state with a symmetry Z a along the torus 73 :…”

Section: S and T Matrices From F Symbolsmentioning

confidence: 99%

“…The T was is given in eq.59 and the basis change B is given by the combination of F-symbols, as shown in 73 : For non-Abelian anyon models, the inversion P −1 for 2 or more dimensional idempotents actually means the sum of inverted simple idempotents. Unlike the matrix of central idempotent, the matrix of simple idempotents P simple in most cases is square and invertible.…”

Section: S and T Matrices From F Symbolsmentioning

confidence: 99%

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Variational methods for characterizing matrix product operator symmetries

Francuz,

Lootens,

Verstraete

et al. 2021

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We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F -symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the F -symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual-and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of Z3, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3 and Rep(S3) string net models.

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Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation

Cited by 42 publications

References 29 publications

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Galois conjugates of String-net Model

Variational methods for characterizing matrix product operator symmetries

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