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arXiv:2101.00426

A $H^{3}(G,{\mathbb T})$-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Yoshiko Ogata

Published Jan 2, 2021• 6 citations
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Abstract

We consider SPT-phases with on-site finite group $G$ symmetry $β$ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$-valued invariant.

DOI
10.48550/arxiv.2101.00426

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Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2×H1(G,Z2)×H2(G,U(1)p)\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})Z2​×H1(G,Z2​)×H2(G,U(1)p​) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.
Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry GGG can be completely classified by the group cohomology class H3(G,U(1))H^3(G, \mathrm{U}(1))H3(G,U(1)). Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of GGG-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional H1(G,Z2)H^1(G, \mathbb{Z}_2)H1(G,Z2​) structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group G×Z2fG\times\mathbb{Z}_2^fG×Z2f​ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.
A method of using partial symmetries to distinguish two dimensional symmetry protected topological (SPT) phases of on-site, unitary symmetries is proposed. This novel order parameter takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, without the need for flux insertion. The construction exploits the rotational symmetry of the lattice to extract on-site SPT invariants, building upon prior work on probing crystalline SPT phases with partial rotations. The method is demonstrated by computing the order parameter analytically on group cohomology models and numerically on a family of states interpolating between the CZX state and a trivial state. Its robustness is suggested by interpreting partial symmetries as generating the topological partition functions of lens spaces.
Abstract We refine our notion of a topoogical phase of matter to account for symmetry protection. We describe SPT (symmetry protected topological or trivial) phases and give some history of this idea. We focus on the case of on-site symmetries. The case of one-dimensional SPTs with on-site symmetry is discussed. For understanding two-dimensional SPTs we focus on the case of Z2 symmetry and argue that there are two possible Z2 paramagnetic phases, i.e., gapped phases where the Z2 symmetry is unbroken. These two phases are in one-to-one correspondence with the two possible Z2 loop gases. We argue that with any discrete group symmetry, we generally classify SPT phases with the possible 3-cocyles. We briefly discuss the idea of symmetry defects and explain that such particles are the excitations of the corresponding twisted Kitaev quantum double model of the same cocyle. We briefly discuss the edge modes of SPT phases. Finally we argue that the SPT phases can be considered as the “ungauging” of the corresponding Dijkgraaf-Witten model.