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arXiv:quant-ph/0303160

$Z_3$-graded Symmetries in Quantum Mechanics

Keivan Aghababaei Samani, Davood Pooladsaz

Published Mar 26, 2003• 0 citations
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Abstract

In this paper we consider $Z_3$-graded topological symmetries (TSs) in one dimensional quantum mechanics. We give a classification of one dimensional quantum systems possessing these symmetries and show that different classes correspond to a positive integer $N$.

DOI
10.48550/arxiv.quant-ph/0303160

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Z3Z_3Z3​-graded Symmetries in Quantum Mechanics
In this paper we consider Z3Z_3Z3​-graded topological symmetries (TSs) in one dimensional quantum mechanics. We give a classification of one dimensional quantum systems possessing these symmetries and show that different classes correspond to a positive integer NNN.
A new class of (2 + 1)-dimensional topological superconductors with Z8\mathbb {Z}_8Z8​ topological classification
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We consider symmetry-protected topological (SPT) phases in two dimensions protected by linear subsystem symmetries, i.e., those that act along rigid lines. There is a distinction between a ``strong'' subsystem SPT phase and a ``weak'' one, which is composed of decoupled one-dimensional SPTs with global symmetries. We propose a natural definition for strong equivalence of such phases, in terms of a linearly symmetric local unitary transformation, under which a weak subsystem SPT is equivalent to the trivial phase. This leads to a number of distinct equivalence classes of strong subsystem SPTs, which we show are in one-to-one correspondence with elements of the group C[Gs]=H2[Gs2,U(1)]/H2[Gs,U(1)]3\mathcal{C}[{G}_{s}]={\mathcal{H}}^{2}[{G}_{s}^{2},U(1)]/{{{\mathcal{H}}^{2}[{G}_{s},U(1)]}}^{3}C[Gs​]=H2[Gs2​,U(1)]/H2[Gs​,U(1)]3, where Gs{G}_{s}Gs​ is the finite Abelian on-site symmetry group. We also show that strong subsystem SPTs by our classification necessarily exhibit a spurious topological entanglement entropy on a cylinder.
Large classes of matter have been systematized before by using a pair of groups to classify all spontaneous symmetry breaking phases or by using a pair of groups plus projective representations to classify all gapped quantum phases of one-dimensional boson and fermion systems with various symmetries. In this paper, the authors go beyond this and use a pair of groups plus three braided fusion categories and a chiral central charge in order to classify gapped quantum phases of two-dimensional boson and fermion systems with various finite symmetries. This allows, among other things, to generate many group tables of two-dimensional gapped phases.