Classification and Construction of Topological Phases of Quantum Matter

TL;DR

This paper introduces a comprehensive theoretical framework for classifying and constructing symmetry protected topological (SPT) phases of quantum matter, unifying various existing proposals and applying algebraic topology tools.

Contribution

It unifies multiple classification schemes for SPT phases into a single framework based on algebraic topology, enabling systematic classification and construction of complex phases.

Findings

Classified 3D fermionic SPT phases with glide symmetry

Classified all 230 space group bosonic SPT phases

Derived a Mayer-Vietoris sequence for reflection symmetry classification

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 $\Omega$-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.