Fermionic symmetry protected topological phases and cobordisms

By A. Kapustin, Ryan Thorngren, Alex Turzillo, Zitao Wang. Journal of High Energy Physics, 2014

358 citations recorded by the source corpus.

It has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} symmetry, where ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known Fermionic SPT phases in space dimension D ≤ 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/b18a23ee3c2f29b9474531a4b89d9f70084c01e1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. E. Witten
2. N. Seiberg
3. Ryan Thorngren
4. D. Gaiotto
5. Chong Wang
6. R. Ma
7. R. Verresen
8. Juven C. Wang

In Condensed Matter Physics

1. E. Witten
2. Ryan Thorngren
3. D. Gaiotto
4. N. Seiberg
5. R. Verresen
6. D. Else
7. R. Ma
8. Juven C. Wang

Citations in the corpus

• The "parity" anomaly on an unorientable manifold (2016, background): “One can think of this factor, which was suggested in [8], as coming from integrating out ν bands4 of gapped bulk fermions [7, 9].”
• Symmetry protected topological phases and generalized cohomology (2017, methodology): “(12)Parts of the paper [28] use just the torsion part of the homotopy groups of ΣIZMSpin; as observed in [11, 28], the non-torsion part seems to describe phases of matter that do not correspond to trulytopological field theories.”
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions (2017, neutral): “Soon after their classification in 1+1D, bosonic SPT states in higher dimensions were also classified based on group cohomology Hd+1(G,U(1)) and Hd+1(G× SO(∞), U(1)) [14–17], 1 or based on cobordism theory [18–20].”
• Orbifold groupoids (2020, background): “Another crucial point is that the world of spin-QFTs includes several interesting invertible theories: besides U(1) phases in d = 0 one has Grassmann-odd one-dimensional vector spaces in d = 1 and the Majorana chain/Arf-invariant theory in d = 2, which assigns partition function (−1)Arf[η] to a mani”
• Dai-Freed anomalies in particle physics (2018, methodology): “A similar table appears in [3].”
• Spin TQFTs and fermionic phases of matter (2015, background): “Some checks of this were performed in [12].”
• The Swampland: Introduction and Review (2019, background): “This leads then to additional constraints on the theory [386–388].”
• Generalized global symmetries (2014, background): “This approach can be generalized to SPT phases with time-reversing symmetries and fermionic degrees of freedom [28,29] (see [30] and [31] for alternative proposals).”
• Average Symmetry-Protected Topological Phases (2022, background): “It is known that fermionic SPT phases protected by U(1) symmetry are classified by the spin cobordism group of a point, Ωspinc(pt) [39, 41, 59, 60].”
• Fermion Path Integrals And Topological Phases (2015, background): “(This was argued in [30] based on an assumption of cobordism invariance.”
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter (2016, methodology): “As an application of these ideas, we propose an explicit construction for all the fermionic SPT phases which are predicted by the spin-cobordism groups [15].”
• JT gravity and the ensembles of random matrix theory (2019, background): “For a more abstract explanation based on cobordism invariance, see [80].”
• Gauging spatial symmetries and the classification of topological crystalline phases (2016, methodology): “We say “bosonic” because we have learned the importance of including spin structure in a careful way [27].”
• Gapless Topological Phases and Symmetry-Enriched Quantum Criticality (2019, background): “[7] Anton Kapustin, Ryan Thorngren, Alex Turzillo, and Zitao Wang, “Fermionic symmetry protected topological phases and cobordisms,” Journal of High Energy Physics 2015, 1–21 (2015).”
• Symmetries and strings of adjoint QCD2 (2020, background): “In 1+1d there are two invertible spin-TQFTs, the trivial theory and the Kitaev chain whose partition function is given by the Arf invariant of the spin structure [36, 89, 93, 97–99].”
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions (2016, methodology): “In addition, we may also understand the spin bordism group Ω 3 (BZ2) ∼= ❩8 classification [84] of those fSPT in terms of three layers with three distinct ❩2 generators respectively through the extended group super-cohomology approach [81–83].”