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Symmetry protected topological phases and generalized cohomology

A preprint version of the article is available at arXiv.


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.


  1. M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1989) 175 [INSPIRE].

    Article  MATH  Google Scholar 

  2. F.J. Burnell, X. Chen, L. Fidkowski and A. Vishwanath, Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order, Phys. Rev. B 90 (2014) 245122 [arXiv:1302.7072] [INSPIRE].

    ADS  Article  Google Scholar 

  3. L. Bhardwaj, D. Gaiotto and A. Kapustin, State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter, JHEP 04 (2017) 096 [arXiv:1605.01640] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. X. Chen, L. Fidkowski and A. Vishwanath, Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator, Phys. Rev. B 89 (2014) 165132 [arXiv:1306.3250] [INSPIRE].

    ADS  Article  Google Scholar 

  5. X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].

    ADS  Article  Google Scholar 

  6. X. Chen, Z.-C. Gu and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization and topological order, Phys. Rev. B 82 (2010) 155138 [arXiv:1004.3835] [INSPIRE].

    ADS  Article  Google Scholar 

  7. X. Chen, Z.-C. Gu and X.-G. Wen, Complete classification of one-dimensional gapped quantum phases in interacting spin systems, Phys. Rev. B 84 (2011) 235128 [arXiv:1103.3323].

    ADS  Article  Google Scholar 

  8. X. Chen, Y.-M. Lu and A. Vishwanath, Symmetry-protected topological phases from decorated domain walls, Nature Commun. 5 (2014) 3507 [arXiv:1303.4301].

    Article  ADS  Google Scholar 

  9. X. Chen, Z.-X. Liu and X.-G. Wen, Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations, Phys. Rev. B 84 (2011) 235141 [arXiv:1106.4752] [INSPIRE].

    ADS  Article  Google Scholar 

  10. L. Fidkowski, X. Chen and A. Vishwanath, Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model, Phys. Rev. X 3 (2013) 041016 [arXiv:1305.5851] [INSPIRE].

    Article  Google Scholar 

  11. D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527 [INSPIRE].

  12. L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83 (2011) 075103 [arXiv:1008.4138].

    ADS  Article  Google Scholar 

  13. D.S. Freed, Short-range entanglement and invertible field theories, arXiv:1406.7278 [INSPIRE].

  14. D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. Z.-C. Gu and M. Levin, Effect of interactions on two-dimensional fermionic symmetry-protected topological phases with Z 2 symmetry, Phys. Rev. B 89 (2014) 201113 [arXiv:1304.4569].

    ADS  Article  Google Scholar 

  16. Z.-C. Gu and X.-G. Wen, Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order, Phys. Rev. B 80 (2009) 155131 [arXiv:0903.1069] [INSPIRE].

    ADS  Article  Google Scholar 

  17. Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev. B 90 (2014) 115141 [arXiv:1201.2648] [INSPIRE].

    ADS  Article  Google Scholar 

  18. L.-Y. Hung and X.-G. Wen, Universal symmetry-protected topological invariants for symmetry-protected topological states, Phys. Rev. B 89 (2014) 075121 [arXiv:1311.5539] [INSPIRE].

    ADS  Article  Google Scholar 

  19. A. Kapustin, Symmetry Protected Topological Phases, Anomalies and Cobordisms: Beyond Group Cohomology, arXiv:1403.1467 [INSPIRE].

  20. A. Kitaev, Homotopy-theoretic approach to spt phases in action: Z 16 classification of three-dimensional superconductors, in Symmetry and Topology in Quantum Matter Workshop, Institute for Pure and Applied Mathematics, University of California, Los Angeles, California (2015) [].

  21. A.Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44 (2001) 131.

    ADS  Article  Google Scholar 

  22. A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134 (2009) 22 [arXiv:0901.2686] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  23. A. Kitaev, On the classification of short-range entangled states, talk at Simons Center, 20 June 2013 [].

  24. A. Kock, L. Kristensen and I. Madsen, Cochain functors for general cohomology theories. I, Math. Scand. 20 (1967) 131.

    MathSciNet  Article  MATH  Google Scholar 

  25. A. Kock, L. Kristensen and I. Madsen, Cochain functors for general cohomology theories. II, Math. Scand. 20 (1967) 151.

    MathSciNet  Article  MATH  Google Scholar 

  26. A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].

  27. A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, JHEP 10 (2017) 080 [arXiv:1701.08264] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  29. F. Pollmann, E. Berg, A.M. Turner and M. Oshikawa. Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85 (2012) 075125 [arXiv:0909.4059].

    ADS  Article  Google Scholar 

  30. G. Segal, The definition of conformal field theory, in Topology, geometry and quantum field theory, Lond. Math. Soc. Lect. Note Ser. 308 (2004) 421.

  31. C. Schommer-Pries, Tori Detect Invertibility of Topological Field Theories, Geom. Topol. 22 (2018) 2713 [arXiv:1511.01772] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  32. A. Vishwanath and T. Senthil, Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect, Phys. Rev. X 3 (2013) 011016 [arXiv:1209.3058] [INSPIRE].

    Article  Google Scholar 

  33. X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88 (2013) 045013 [arXiv:1303.1803] [INSPIRE].

    ADS  Google Scholar 

  34. X.-G. Wen, Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions, Phys. Rev. B 89 (2014) 035147 [arXiv:1301.7675] [INSPIRE].

    ADS  Article  Google Scholar 

  35. Q.-R. Wang and Z.-C. Gu, Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory, Phys. Rev. X 8 (2018) 011055 [arXiv:1703.10937] [INSPIRE].

    Article  Google Scholar 

  36. C. Wang, C.-H. Lin and Z.-C. Gu, Interacting fermionic symmetry-protected topological phases in two dimensions, Phys. Rev. B 95 (2017) 195147 [arXiv:1610.08478] [INSPIRE].

    ADS  Article  Google Scholar 

  37. C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B 89 (2014) 195124 [Erratum ibid. B 91 (2015) 239902] [arXiv:1401.1142] [INSPIRE].

  38. C.Z. Xiong, Minimalist approach to the classification of symmetry protected topological phases, J. Phys. A 51 (2018) 445001 [arXiv:1701.00004] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

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Correspondence to Davide Gaiotto.

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ArXiv ePrint: 1712.07950

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Gaiotto, D., Johnson-Freyd, T. Symmetry protected topological phases and generalized cohomology. J. High Energ. Phys. 2019, 7 (2019).

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  • Effective Field Theories
  • Global Symmetries
  • Topological Field Theories
  • Topological States of Matter