Skip to main content
SpringerLink
Log in

Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains Without Continuous Symmetry

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

Abstract

We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses time-reversal symmetry or \({\mathbb{Z}_{2} \times \mathbb{Z}_{2}}\) symmetry (i.e., the symmetry with respect to the \({\pi}\) rotations of spins about the three orthogonal axes). The proof is based on the deep analogy between the matrix product state formulation and the representation of the Cuntz algebra in the von Neumann algebra \({\pi(\mathcal{A}_{R})''}\) constructed from the ground state restricted to the right half-infinite chain.

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

Similar content being viewed by others

References

  1. Affleck I., Lieb E.H.: A proof of part of Haldane’s conjecture on spin chains. Lett. Math. Phys. 12, 57–69 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994) https://projecteuclid.org/euclid.cmp/1104270709

    Article  ADS  MathSciNet  Google Scholar 

  3. Arveson, W.B.: Continuous Analogues of Fock space I, vol. 409. Memoirs of the American Mathematical Society, Providence (1989)

    Article  MathSciNet  Google Scholar 

  4. Bratteli O., Jorgensen P., Kishimoto A., Werner R.F.: Pure states on \({\mathcal{O}_d}\). J. Oper. Theory 43, 97–143 (2000)

    Google Scholar 

  5. Bratteli, O., Jorgensen, P., Price, G.: Endomorphisms of \({B(\mathcal{H})}\). Quantization, nonlinear partial differential equations, and operator algebra, pp. 93–138. In: Proceedings of Symposia in Pure Mathematics, vol. 59 (1996) https://www.duo.uio.no/handle/10852/43152

  6. Bratteli O., Jorgensen P.E.T.: Endomorphisms of B(H) II. Finitely correlated states on O n. J. Funct. Anal. 145, 323–373 (1997)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

  9. Chen X., Gu Z.-C., Wen X.-G.: Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B. 83, 035107 (2011) arXiv:1008.3745

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  11. Fannes, M., Nachtergaele, B., Werner, R. F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992) https://projecteuclid.org/euclid.cmp/1104249404

    Article  ADS  MathSciNet  Google Scholar 

  12. Hastings, M.: An area law for one-dimensional quantum systems. J. Stat. Mech. P08024 (2007) arXiv:0705.2024

  13. Hastings M.B.: Lieb–Schultz–Mattis in higher dimensions. Phys. Rev. B. 69, 104431 (2004) arXiv:1001.5280

    Article  ADS  Google Scholar 

  14. Hastings M.B.: Sufficient conditions for topological order in insulators. Eur. Phys. Lett. 70, 824–830 (2005) arXiv:cond-mat/0411094

    Article  ADS  Google Scholar 

  15. Lieb E., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  16. Matsui T.: A characterization of finitely correlated pure states. Infinite Dimens. Anal. Quantum Probab. 1, 647–661 (1998)

    Article  Google Scholar 

  17. Matsui T.: The split property and the symmetry breaking of the quantum spin chain. Commun. Math. Phys. 218, 393–416 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  18. Matsui, T.: Boundedness of entanglement entropy and split property of quantum spin chains. Rev. Math. Phys. 1350017 (2013). arXiv:1109.5778

  19. Nachtergaele, B.: Private communication

  20. Nachtergaele B., Sims R.: A multi-dimensional Lieb–Schultz–Mattis theorem. Commun. Math. Phys. 276, 437–472 (2007) arXiv:math-ph/0608046

    Article  ADS  MathSciNet  Google Scholar 

  21. Oshikawa M.: Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice. Phys. Rev. Lett. 84, 1535 (2000) arXiv:cond-mat/9911137

    Article  ADS  Google Scholar 

  22. Oshikawa M., Yamanaka M., Affleck I.: Magnetization plateaus in spin chains: “Haldane gap” for half-integer spins. Phys. Rev. Lett. 78, 1984 (1997) arXiv:cond-mat/9610168

    Article  ADS  Google Scholar 

  23. Parameswaran S.A., Turner A.M., Arovas D.P., Vishwanath A.: Topological order and absence of band insulators at integer filling in non-symmorphic crystals. Nat. Phys. 9, 299–303 (2013) arXiv:1212.0557

    Article  Google Scholar 

  24. 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, 167202 (2008) arXiv:0802.0447

    Article  ADS  Google Scholar 

  25. Pollmann F., Turner A.M., Berg E., Oshikawa M.: Entanglement spectrum of a topological phase in one dimension. Phys. Rev. B. 81, 064439 (2010) arXiv:0910.1811

    Article  ADS  Google Scholar 

  26. Sutherland B.: Beautiful Models—70 Years of Exactly Solved Quantum Many-Body Problems. World Scientific, Singapore (2004)

    Book  Google Scholar 

  27. Takesaki M.: Theory of Operator Algebras. I. Encyclopaedia of Mathematical Sciences. Springer, Berlin (2002)

    MATH  Google Scholar 

  28. Tasaki H.: Lieb–Schultz–Mattis theorem with a local twist for general one-dimensional quantum systems. J. Stat. Phys. 170, 653–671 (2018) arXiv:1708.05186

    Article  ADS  MathSciNet  Google Scholar 

  29. Tasaki, H.: Physics and Mathematics of Quantum Many-Body Systems (to be published from Springer)

  30. Yamanaka M., Oshikawa M., Affleck I.: Nonperturbative approach to Luttinger’s theorem in one dimension. Phys. Rev. Lett. 79, 1110 (1997) arXiv:cond-mat/9701141

    Article  ADS  Google Scholar 

  31. Watanabe H.: The Lieb–Schultz–Mattis-type filling constraints in the 1651 magnetic space groups. Phys. Rev. B. 97, 165117 (2018) arXiv:1802.00587

    Article  ADS  Google Scholar 

  32. Watanabe, H., Po, H.C., Vishwanath, A., Zaletel, M.P.: Filling constraints for spin–orbit coupled insulators in symmorphic and nonsymmorphic crystals. Proc. Natl. Acad. Sci. USA 112, 14551–14556 (2015) http://www.pnas.org/content/112/47/14551.short

    Article  ADS  Google Scholar 

  33. Zeng, B., Chen, X., Zhou, D.-L., Wen, X.-G.: Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phase in Many-Body Systems (to be published from Springer) arXiv:1508.02595

Download references

Acknowledgements

It is a pleasure to thank HarukiWatanabe for valuable discussion which was essential for the present work, and Tohru Koma for useful discussion and comments. We also thank TakuMatsui and Bruno Nachtergaele for useful comments. The present work was supported by JSPS Grants-in-Aid for Scientific Research nos. 16K05171 (Y.O.) and 16H02211 (H.T.).

Author information

Authors and Affiliations

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

    Yoshiko Ogata

  2. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

    Hal Tasaki

Authors
  1. Yoshiko Ogata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hal Tasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshiko Ogata.

Additional information

Communicated by H.-T. Yau

Publisher’s Note

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

Yoshiko 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., Tasaki, H. Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains Without Continuous Symmetry. Commun. Math. Phys. 372, 951–962 (2019). https://doi.org/10.1007/s00220-019-03343-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03343-5

Search

Navigation