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arXiv:1602.08839

C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry

Yoshiko Ogata

Published Feb 29, 2016• 1 citation
arXivHTMLPDF

Abstract

We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains $\bigotimes_{\mathbb Z}\mathop{\mathrm{M}}\nolimits_n$, with local symmetry given by a group $G$. The $G$-symmetric gapped Hamiltonians are given by triples $(k,u,V)$, where $u$ is a projective unitary representation of $G$ on a finite dimensional space ${\mathbb C}^k$, and $V$ is an isometry from ${\mathbb C}^k$ to ${\mathbb C}^n\otimes {\mathbb C}^k$. We show that Hamiltonians $H_0,H_1$, given by the triples $(k^{(0)}, u^{(0)}, V^{(0)})$ and $(k^{(1)}, u^{(1)}, V^{(1)})$ are $C^1$-equivalent if the projective representations $u^{(0)}$ and $u^{(1)}$ are unitary equivalent.

DOI
10.48550/arxiv.1602.08839

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Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for strongly interacting systems. Here we make an attempt to answer these questions for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range-correlated states in the same phase, we classify possible quantum phases for one-dimensional (1D) matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if a certain symmetry is required, many phases exist with different symmetry-protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.
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Projective symmetry groups beyond spin-1/21/21/2: Classifying integer-spin quantum spin liquids
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Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.