Symmetry protected topological phases and generalized cohomology
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Springer
Received: September 26, Revised: March 25, Accepted: April 22, Published: May 2,
2018 2019 2019 2019
Davide Gaiotto and Theo Johnson-Freyd Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada
E-mail: [email protected], [email protected] 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. Keywords: Effective Field Theories, Global Symmetries, Topological Field Theories, Topological States of Matter ArXiv ePrint: 1712.07950
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)007
JHEP05(2019)007
Symmetry protected topological phases and generalized cohomology
Contents 1 3
2 Generalities 2.1 Stacking and invertibility 2.2 SREs, invertible phases of matter, and invertible TFTs 2.3 Defects 2.4 Invertible defects
3 3 4 5 6
3 The spectrum of invertible phases 8 3.1 Definition of the spectrum 8 × 3.2 Cohomology with coefficients in GP and internal symmetries in lattice models 11 4 SPT phases and spectra 4.1 Symmetries and defects 4.2 Stable cohomology operations 4.3 SPT phases and reduced cohomology 4.4 Phases versus choices of realization 4.5 Stacking of SPT phases
12 12 16 17 17 17
5 Examples 5.1 Bosonic SPT phases and standard group cohomology 5.2 Bosonic SPT phases beyond group cohomology 5.3 Restricted supercohomology 5.4 Extended supercohomology 5.5 Algebraic description of extended supercohomology 5.6 Beyond the Majorana layer 5.7 A spectrum of SPT phases
18 18 19 20 21 23 23 24
6 Categorical actions and antiunitarity 6.1 Time-reversal and ordinary group cohomology 6.2 Time reversal and E8 6.3 Restricted Gu-Wen phases with time reversal 6.4 Extended supercohomology with time reversal
24 25 26 27 28
7 SPT phases as anomaly theories
30
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JHEP05(2019)007
1 Introduction and conclusions 1.1 Outline of the paper
1
Introduction and conclusions
• SPT phases in d + 1 spacetime dimensions are classified by a reduced generalized e d+1 (BGb ; GP× ), there Gb is the bosonic symmetry of the cohomology group H ≤d+1 theory, i.e. the quotient of the on-site internal symmetry group by fermion parity operation. × • • The target GP× ≤d+1 of the generalized cohomology theory H (−; GP≤d+1 ) is essentially the same as the space of invertible phases of matter in spacetime-dimension 1
See [38] for an earlier discussion of Kitaev’s proposal and its consequences.
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JHEP05(2019)007
The general classification of Symmetry Protected Topological phases of matter is an important problem in theoretical physics and mathematics [2, 4–7, 9, 11, 12, 16–19, 21– 23, 28, 29, 32–34, 37, 38]. The typical approach to the problem in the condensed mat
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