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arXiv:1712.07950 · Journal of High Energy Physics

Symmetry protected topological phases and generalized cohomology

Davide Gaiotto A, Theo Johnson-Freyd

Published May 1, 2019• 141 citations
arXivHTMLPDF

Abstract

Abstract We discuss the classification of SPT phases in condensed matter systems. We review Kitaev’s argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems [20, 23]. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.

DOI
10.1007/jhep05(2019)007

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Abstract We discuss the classification of SPT phases in condensed matter systems. We review Kitaev’s argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems [20, 23]. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.
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 ΩΩΩ-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.
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 the 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.
Abstract A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist generalized cohomology hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3D bosonic SPT phases with space group symmetry G is , where the H 1 term classifies phases beyond the Borel group cohomology proposal.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
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.