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Title: Fermionic symmetry protected topological phases and cobordisms

Abstract

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 \( {\mathrm{\mathbb{Z}}}_2 \) symmetry, where \( {\mathrm{\mathbb{Z}}}_2 \) 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

Authors:
 [1];  [2];  [1];  [1]
  1. California Inst. of Technology (CalTech), Pasadena, CA (United States)
  2. Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
California Institute of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1599360
Grant/Contract Number:  
SC0011632; FG02-92ER40701
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 12; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Topological Field Theories; Effective field theories; Topological States of Matter

Citation Formats

Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, and Wang, Zitao. Fermionic symmetry protected topological phases and cobordisms. United States: N. p., 2015. Web. https://doi.org/10.1007/JHEP12(2015)052.
Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, & Wang, Zitao. Fermionic symmetry protected topological phases and cobordisms. United States. https://doi.org/10.1007/JHEP12(2015)052
Kapustin, Anton, Thorngren, Ryan, Turzillo, Alex, and Wang, Zitao. Thu . "Fermionic symmetry protected topological phases and cobordisms". United States. https://doi.org/10.1007/JHEP12(2015)052. https://www.osti.gov/servlets/purl/1599360.
@article{osti_1599360,
title = {Fermionic symmetry protected topological phases and cobordisms},
author = {Kapustin, Anton and Thorngren, Ryan and Turzillo, Alex and Wang, Zitao},
abstractNote = {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 \( {\mathrm{\mathbb{Z}}}_2 \) symmetry, where \( {\mathrm{\mathbb{Z}}}_2 \) 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},
doi = {10.1007/JHEP12(2015)052},
journal = {Journal of High Energy Physics (Online)},
number = 12,
volume = 2015,
place = {United States},
year = {2015},
month = {12}
}

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Cited by: 89 works
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Works referenced in this record:

Exotic structures on 4-manifolds detected by spectral invariants
journal, February 1988


The Structure of the Spin Cobordism Ring
journal, September 1967

  • Anderson, D. W.; Brown, E. H.; Peterson, F. P.
  • The Annals of Mathematics, Vol. 86, Issue 2
  • DOI: 10.2307/1970690

Dimers on Surface Graphs and Spin Structures. I
journal, July 2007

  • Cimasoni, David; Reshetikhin, Nicolai
  • Communications in Mathematical Physics, Vol. 275, Issue 1
  • DOI: 10.1007/s00220-007-0302-7

A calculation ofPin + bordism groups
journal, December 1990

  • Kirby, R. C.; Taylor, L. R.
  • Commentarii Mathematici Helvetici, Vol. 65, Issue 1
  • DOI: 10.1007/BF02566617

Pin cobordism and related topics
journal, December 1969

  • Anderson, D. W.; Brown, E. H.; Peterson, F. P.
  • Commentarii Mathematici Helvetici, Vol. 44, Issue 1
  • DOI: 10.1007/BF02564545

Interacting fermionic topological insulators/superconductors in three dimensions
journal, May 2014


Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model
journal, November 2013


Periodic table for topological insulators and superconductors
conference, January 2009

  • Kitaev, Alexei; Lebedev, Vladimir; Feigel’man, Mikhail
  • ADVANCES IN THEORETICAL PHYSICS: Landau Memorial Conference, AIP Conference Proceedings
  • DOI: 10.1063/1.3149495

Symmetry-protected topological phases from decorated domain walls
journal, March 2014

  • Chen, Xie; Lu, Yuan-Ming; Vishwanath, Ashvin
  • Nature Communications, Vol. 5, Issue 1
  • DOI: 10.1038/ncomms4507

Generalizations of the Kervaire Invariant
journal, March 1972

  • Brown, Edgar H.
  • The Annals of Mathematics, Vol. 95, Issue 2
  • DOI: 10.2307/1970804

Symmetry protected topological orders and the group cohomology of their symmetry group
journal, April 2013


Topological phases of fermions in one dimension
journal, February 2011


Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010


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