Dyson’s classification and real division superalgebras

Geiko, R.V.; Moore, Gregory W.

doi:10.1007/jhep04(2021)299

Cited by 3 publications

(1 citation statement)

References 15 publications

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“…The simplest superalgebras are the superdivision algebras: superalgebras in which every homogeneous element is invertible. Algebraically, the origin of the tenfold way arises from the following super-version of the Frobenius theorem on real division algebras [11].…”

Section: The Group Of Spt Phasesmentioning

confidence: 99%

Interacting SPT phases are not morita invariant

Stehouwer

2022

Lett Math Phys

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The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with K-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by K-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita equivalent but different symmetry groups. As K-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.

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Section: The Group Of Spt Phasesmentioning

confidence: 99%

Interacting SPT phases are not morita invariant

Stehouwer

2022

Lett Math Phys

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When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

Geiko

Moore

2023

Ann. Henri Poincaré

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Non-local order parameters for fermion chains via the partial transpose

Mayer

2023

Journal of Mathematical Physics

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In the last two decades, a vast variety of topological phases have been described, predicted, classified, proposed, and measured. While there is a certain unity in method and philosophy, the phenomenology differs wildly. This work deals with the simplest such case: fermions in one spatial dimension, in the presence of a symmetry group G, which contains anti-unitary symmetries. A complete classification of topological phases, in this case, is available. Nevertheless, these methods are to some extent lacking as they generally do not allow to determine the class of a given system easily. This paper will take up proposals for non-local order parameters defined through anti-unitary symmetries. They are shown to be homotopy invariants on a suitable set of ground states. For matrix product states, an interpretation of these invariants is provided: in particular, for a particle–hole symmetry, the invariant determines a real division super algebra [Formula: see text] such that the bond algebra is a matrix algebra over [Formula: see text].

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Dyson’s classification and real division superalgebras

Cited by 3 publications

References 15 publications

Interacting SPT phases are not morita invariant

Interacting SPT phases are not morita invariant

When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

Non-local order parameters for fermion chains via the partial transpose

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