Skip to main content
SpringerLink
Log in

A \({{\mathbb {Z}}}_2\)-index of Symmetry Protected Topological Phases with Reflection Symmetry for Quantum Spin Chains

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

For the classification of SPT phases, defining an index is a central problem. In the famous paper  (Pollmann et al. Phys Rev B 81:064439, 2010), Pollmann, Tuner, Berg, and Oshikawa introduced \({{\mathbb {Z}}}_2\)-indices for injective matrix products states which have either \({{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2\) dihedral group (of \(\pi \)-rotations about x, y, and z-axes) symmetry, time-reversal symmetry, or reflection symmetry. The first two are on-site symmetries. In Ogata in A \({\mathbb {Z}}_2\)-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains. arXiv:1810.01045, an index for on-site symmetries, which generalizes the index in Pollmann et al. (2010), was introduced for general unique gapped ground state phases in quantum spin chains. It was proved that the index is an invariant of the \(C^1\)-classification of SPT phases. The index for the reflection symmetry, which is not an on-site symmetry, was left as an open question. In this paper, we introduce a \({{\mathbb {Z}}}_2\)-index for the reflection symmetric unique gapped ground state phases, and complete the generalization problem of index by Pollmann et al. We also show that the index is an invariant of the \(C^1\)-classification.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Comm. Math. Phys. 115, 477–528 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  2. Arveson, W.B.: Continuous analogues of Fock space I. Mem. Amer. Math. Soc. 409, (1989)

  3. Bachmann, S., Michalakis, S., Nachtergaele, B., Sims, R.: Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Communications in Mathematical Physics 309, 835–871 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  4. Bachmann, S., Nachtergaele, B.: On gapped phases with a continuous symmetry and boundary operators. J. Stat. Phy. 154, 91–112 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bratteli, O., Jorgensen, P., Price, G.: Endomorphisms of \(B({\cal{H}})\). Quantization, nonlinear partial differential equations, and operator algebra. 93–138, Proc. Sympos. Pure Math., 59, (1996)

  6. Bratteli, O., Jorgensen, P.E.T.: Endomorphisms of \(B(H)\) II. Finitely Correlated States on \(O_n\). Journal of functional analysis. 145, 323–373 (1997)

    Article  MathSciNet  Google Scholar 

  7. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer-Verlag, Berlin (1986)

    MATH  Google Scholar 

  8. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer-Verlag, Berlin (1996)

    MATH  Google Scholar 

  9. Derezinski, J., Jaksic, V., Pillet, C.-A.: Perturbation theory of \(W^*\)-dynamics, Liouvilleans and KMS-states. Reviews in Mathematical Physics 15–05, 447–489 (2003)

    Article  MathSciNet  Google Scholar 

  10. Gu, Z.-C., Wen, X.-G.: Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B 80, (2009)

  11. Chen, X., Gu, Z.-C., Wen, X.-G.: Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B 83, (2011)

  12. Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  13. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states on quantum spin chains. Comm. Math. Phys. 144, 443–490 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  14. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated pure states. Journal of functional analysis. 120, 511–534 (1994)

    Article  MathSciNet  Google Scholar 

  15. Haldane, F.D.M.: Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the \(O(3)\) nonlinear sigma model. Phys. Lett. 93A, 464–468 (1983)

    Article  MathSciNet  Google Scholar 

  16. Haldane, F.D.M.: Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  17. Hastings, M.: An area law for one-dimensional quantum systems. Journal of Statistical Mechanics. P08024, (2007)

  18. Hastings, M.: Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance. (2010) arXiv:1001.5280v2 [math-ph]

  19. Kennedy, T.: Exact diagonalization of open spin 1 chains,. J. Phys.: Cond.Matt. 2, 5737–5745 (1990)

    ADS  Google Scholar 

  20. Koma, T., Nachtergaele, B.: The Spectral Gap of the Ferromagnetic XXZ-Chain. Letters in Mathematical Physics 40, 1–16 (1997)

    Article  MathSciNet  Google Scholar 

  21. Kennedy, T., Tasaki, H.: Hidden \({{\mathbb{Z}}}_2\times {{\mathbb{Z}}}_2\)-symmetry breaking in Haldane-gap antiferromagnets. Phys. Rev. B 45, 304–307 (1992)

    Article  Google Scholar 

  22. Kennedy, T., Tasaki, H.: Hidden symmetry breaking and the Haldane phase in \(S= 1\) quantum spin chains. Communications in Mathematical Physics 147, 431–484 (1992)

    Article  MathSciNet  Google Scholar 

  23. Matsui, T.: A characterization of matrix product pure states. Infinite dimensional analysis and quantum probability. 1, 647–661 (1998)

    MATH  Google Scholar 

  24. Matsui, T.: The split property and the symmetry breaking of the quantum spin chain. Communications in Mathematical Physics 218, 393–416 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  25. Matsui, T.: Boundedness of entanglement entropy and split property of quantum spin chains. Reviews in Mathematical Physics 1350017, (2013)

  26. Nachtergaele, B., Ogata, Y., Sims, R.: Boundedness of entanglement entropy and split property of quantum spin chains. J. Stat. Phys 124, 1–13 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  27. den Nijs, M., Rommelse, K.: Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. Rev. B 40, 4709 (1989)

    Article  ADS  Google Scholar 

  28. Ogata, Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I. Communications in Mathematical Physics 348, 847–895 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  29. Ogata, Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification II. Communications in Mathematical Physics 348, 897–957 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  30. Ogata, Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification III. Communications in Mathematical Physics 352, 1205–1263 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  31. Ogata, Y.: A \({\mathbb{Z}}_2\)-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains arXiv:1810.01045

  32. Ogata, Y., Tasaki, H.: Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry. Communications in Mathematical Physics, (2019)

  33. Pollmann, F., Turner, A., Berg, E., Oshikawa, M.: Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B 81, 064439 (2010)

    Article  ADS  Google Scholar 

  34. Pollmann, F., Turner, A., Berg, E., Oshikawa, M.: Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 81, 075125 (2012)

    Article  ADS  Google Scholar 

  35. Perez-Garcia, D., Wolf, M.M., Sanz, M., Verstraete, F., Cirac, J.I.: String order and symmetries in quantum spin lattices. Phys. Rev. Lett. 100, (2008)

  36. Takesaki, M.: Theory of operator algebras I. Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2002)

    MATH  Google Scholar 

  37. Takesaki, M.: Theory of operator algebras II. Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2003)

    Book  Google Scholar 

  38. Tasaki, H.: Topological phase transition and Z2 index for S = 1 quantum spin chains arXiv:1804.04337

  39. Tasaki, H.: Physics and mathematics of quantum many-body systems, (to be published from Springer)

  40. Wolf, M.M.: Quantum channels & operations. Unpublished. (2012)

Download references

Acknowledgements

The author is grateful to Hal Tasaki for fruitful discussion which was essential for the present work, and for the helpful comments on the manuscript. The beginning of the introduction heavily relies on his help. This work was supported by JSPS KAKENHI Grant No. 16K05171.

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Yoshiko Ogata

Authors
  1. Yoshiko Ogata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshiko Ogata.

Additional information

Communicated by M. Wolf

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Y. Ogata: Supported in part by the Grants-in-Aid for Scientific Research, JSPS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ogata, Y. A \({{\mathbb {Z}}}_2\)-index of Symmetry Protected Topological Phases with Reflection Symmetry for Quantum Spin Chains. Commun. Math. Phys. 385, 1245–1272 (2021). https://doi.org/10.1007/s00220-021-04057-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-04057-3

Search

Navigation