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arXiv · 1906.02892

Classification and Construction of Topological Phases of Quantum Matter

Charles Zhaoxi Xiong
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Abstract

We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on Kitaev's idea that short-range entangled states form what are known as $Ω$-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.

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Charles Zhaoxi Xiong. 2019-06-07. Classification and Construction of Topological Phases of Quantum Matter. https://arxiv.org/abs/1906.02892

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