Classification and Construction of Topological Phases of Quantum Matter
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 -spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.
Keywords
Cite
@article{arxiv.1906.02892,
title = {Classification and Construction of Topological Phases of Quantum Matter},
author = {Charles Zhaoxi Xiong},
journal= {arXiv preprint arXiv:1906.02892},
year = {2019}
}
Comments
PhD thesis in physics submitted to Harvard University, based on prior works arXiv:1701.00004 [single-author, published in J. Phys. A 51 (44):445001 (2018)], arXiv:1709.06998 [joint with A. Alexandradinata, published in Phys. Rev. B 97:115153 (2018)], arXiv:1810.00801 (joint with Ken Shiozaki and Kiyonori Gomi), and arXiv:1811.06558 (joint with Hao Song and Sheng-Jie Huang)