Anomaly indicators for topological orders with 
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>
 and time-reversal symmetry

Lapa, Matthew F.; Levin, Michael

doi:10.1103/physrevb.100.165129

Cited by 11 publications

(24 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now I ′ 2 has the interpretation of the local T 2 value for v in the new SET phase, which is the reference one modified by the fractionalization class t. I 3 is determined by whether the vison is a boson or fermion. These are precisely the known anomaly indicators for U(1) × Z T 2 symmetry [27]. Let us apply these formulas to an example, where we reproduce the results of [30].…”

Section: General Anomaly Formulamentioning

confidence: 76%

“…These diagnostics were used recently in Ref. 27 to derive formulas for absolute anomaly indicators for U(1) × Z T 2 and U(1) ⋊ Z T 2 symmetry groups. Below we apply our relative anomaly formula to compare with these results.…”

Section: A Relation To Absolute Anomaly Indicator Z(rp 4 )mentioning

confidence: 99%

“…Recently, anomaly indicators for U(1) × Z T 2 and U(1) ⋊ Z T 2 were also derived in Ref. 27, allowing a general computation of mixed anomalies between U(1) and Z T 2 symmetry. For general symmetry groups that involve space-time reflections, the only known result is an anomaly formula for Abelian toric code topological order when anyons are not permuted by symmetries, derived from an explicit microscopic construction of SET phases in Ref.…”

mentioning

confidence: 99%

“…We subsequently use the relative anomaly formula to reproduce previous results for Z T 2 , U(1)×Z T 2 , and U(1)⋊Z T 2 anomalies [21,27] in a completely different way, thus providing a highly non-trivial consistency check on the correctness of this approach.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Relative anomalies in (2+1)D symmetry enriched topological states

Barkeshli

Cheng

2020

SciPost Phys.

View full text Add to dashboard Cite

Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for Z T 2 space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for U (1)×Z T 2 and U (1)⋊Z T 2 symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as Z T 4 and mixed anomalies for Z2 × Z T 2 symmetry, and unitary Z2 × Z2 symmetry with non-trivial anyon permutations. CONTENTS

show abstract

Section: General Anomaly Formulamentioning

confidence: 76%

Section: A Relation To Absolute Anomaly Indicator Z(rp 4 )mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Relative anomalies in (2+1)D symmetry enriched topological states

Barkeshli

Cheng

2020

SciPost Phys.

View full text Add to dashboard Cite

show abstract

“…Subsequently, for U (1) × Z T 2 and U (1) Z T 2 symmetry, Ref. [45] recently presented formulas for anomaly indicators that apply to general topological orders and allow anyon permutations.…”

Section: A Relation To Prior Workmentioning

confidence: 99%

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Bulmash

Barkeshli

2020

Phys. Rev. Research

View full text Add to dashboard Cite

Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a G-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D G-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H 4 (G, U (1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date.

show abstract

Numerical identification and gapped boundaries of Abelian fermionic topological order

Bultinck¹

2020

J. Stat. Mech.

View full text Add to dashboard Cite

In this work we consider general fermion systems in two spatial dimensions, both with and without charge conservation symmetry, which realize a non-trivial fermionic topological order with only Abelian anyons. We address the question of precisely how these quantum phases differ from their bosonic counterparts, both in terms of their edge physics and in the way one would identify them in numerics. As in previous works, we answer these questions by studying the theory obtained after gauging the global fermion parity symmetry, which turns out to have a special and simple structure. Using this structure, a minimal scheme is outlined for how to numerically identify a general Abelian fermionic topological order, without making use of fermion number conservation. Along the way, some subtleties of the momentum polarization technique are discussed. Regarding the edge physics, it is shown that the gauged theory can have a (bosonic) gapped boundary to the vacuum if and only if the ungauged fermion theory has a gapped boundary as well.

show abstract

Anomaly indicators for topological orders with U(1) and time-reversal symmetry

Cited by 11 publications

References 66 publications

Relative anomalies in (2+1)D symmetry enriched topological states

Relative anomalies in (2+1)D symmetry enriched topological states

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Numerical identification and gapped boundaries of Abelian fermionic topological order

Contact Info

Product

Resources

About