Anomalous domain wall condensation in a modified Ising chain

Roose, Gertian; Vanderstraeten, Laurens; Haegeman, Jutho; Bultinck, Nick

doi:10.1103/physrevb.99.195132

Cited by 5 publications

(6 citation statements)

References 81 publications

(135 reference statements)

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“…We note that the connection between domain wall F -symbols and anomalies was also alluded to in Ref. 27 in the context of a Z 2 SPT edge theory.…”

Section: Introductionmentioning

confidence: 55%

Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

Kawagoe,

Levin

2021

Preprint

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We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element ω of the cohomology group H 3 (G, UT (1)). This element ω ∈ H 3 (G, UT (1)) can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the F -symbol associated with domain walls in a symmetry broken edge theory; this domain wall F -symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.2 Alternatively, we can break the symmetry explicitly rather than spontaneously. See Sec. VII for details.

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“…We note that the connection between domain wall F -symbols and anomalies was also alluded to in Ref. 27 in the context of a Z 2 SPT edge theory.…”

Section: Introductionmentioning

confidence: 55%

Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

Kawagoe,

Levin

2021

Preprint

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“…A realization of this phase can be found in Ref. [16] with Hamiltonian H = i CZ i,i+2 X i+1 − µZ i Z i+1 for the parameter range µ > 0. The CZ gate acts on two qubits as CZ|ab = (−1) a•b |ab for a, b = {0, 1} corresponding to the Z basis.…”

Section: Mpo Algebras Representing Zmentioning

confidence: 99%

“…Generalized symmetries of (1+1)d systems, also in the form of MPOs, have been studied before for concrete models, revealing novel features not present in standard SPT phases. MPO symmetries of abelian groups give rise to anomalous domain wall excitations [16,17] and topological symmetries based on truncated su(2) deformations studied in anyonic spin chains [18][19][20] protect gapless phases present in those systems. Interestingly, symmetries of non-truncated su (2) deformations have recently been argued to host SPT phases [21,22].…”

Section: Introductionmentioning

confidence: 99%

Classifying phases protected by matrix product operator symmetries using matrix product states

Garre-Rubio

Lootens

Molnár

2023

Quantum

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We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.

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“…A realization of this phase can be found in Ref. [24] with Hamiltonian H = i CZ i,i+2 X i+1 − µZ i Z i+1 for the parameter range µ > 0. This Hamiltonian is invariant under U = CZ i,i+1 Z i X i that is an MPO representation of Z 2 with the non-trivial cocycle.…”

Section: Mpo Algebras Representing Z2mentioning

confidence: 99%

“…MPO symmetries have also been studied in purely (1+1)d systems representing abelian groups [24,25] and su (2) deformations in anyonic spin chains [26][27][28], where the symmetry is called 'topological symmetry'.…”

Section: Introductionmentioning

confidence: 99%

Classifying phases protected by matrix product operator symmetries using matrix product states

Garre-Rubio¹,

Lootens²,

Molnár³

2022

Preprint

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We classify the different ways in which matrix product states (MPS) can transform under the action of matrix product operator (MPO) symmetries. We give a local characterization of MPS tensors that generate ground spaces which remain invariant under a global MPO symmetry and generally correspond to symmetry breaking phases. This allows us to derive a set of quantities, satisfying the so-called coupled pentagon equations, invariant under continuous deformations of the MPS tensor, which provides the invariants to detect different gapped phases protected by MPO symmetries. The results obtained match with previous TQFT approaches classifying non-onsite symmetries, our techniques, however, extend the classification beyond renormalization fixed points and facilitates the numerical study of these systems. When the MPO symmetries form a global onsite representation of a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. When the MPO symmetries form a representation of a group, but this representation is not onsite, we obtain restrictions on the possible ground state degeneracies depending both on the group and on the concrete form of the MPO representation. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and also provide examples of our classification together with explicit constructions of MPO and MPS. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where non-onsite symmetries also play a key role.

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Anomalous domain wall condensation in a modified Ising chain

Cited by 5 publications

References 81 publications

Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

Classifying phases protected by matrix product operator symmetries using matrix product states

Classifying phases protected by matrix product operator symmetries using matrix product states

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