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

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.

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References

  1. AL

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

    ADS  MathSciNet  Article  Google Scholar 

  2. AN

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

    ADS  MathSciNet  Article  Google Scholar 

  3. A

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

    MathSciNet  Article  Google Scholar 

  4. BJKW

    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. BJP

    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. BJ

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

    MathSciNet  Article  Google Scholar 

  7. BR1

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

    Google Scholar 

  8. BR2

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

    Google Scholar 

  9. CGW

    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

    ADS  Article  Google Scholar 

  10. DL

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

    ADS  MathSciNet  Article  Google Scholar 

  11. FNW

    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

    ADS  MathSciNet  Article  Google Scholar 

  12. H1

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

  13. H2

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

    ADS  Article  Google Scholar 

  14. H3

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

    ADS  Article  Google Scholar 

  15. LSM

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

    ADS  MathSciNet  Article  Google Scholar 

  16. M1

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

    Article  Google Scholar 

  17. M2

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

    ADS  MathSciNet  Article  Google Scholar 

  18. M3

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

  19. N

    Nachtergaele, B.: Private communication

  20. NS

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

    ADS  MathSciNet  Article  Google Scholar 

  21. O

    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

    ADS  Article  Google Scholar 

  22. OYA

    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

    ADS  Article  Google Scholar 

  23. PTAV

    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. PWSVC

    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

    ADS  Article  Google Scholar 

  25. PTBO

    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

    ADS  Article  Google Scholar 

  26. S

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

    Google Scholar 

  27. Tak

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

    Google Scholar 

  28. Tas1

    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

    ADS  MathSciNet  Article  Google Scholar 

  29. Tas2

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

  30. YOA

    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

    ADS  Article  Google Scholar 

  31. Wa

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

    ADS  Article  Google Scholar 

  32. WPVZ

    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

    ADS  Article  Google Scholar 

  33. ZCZW

    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

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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.).

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Correspondence to Yoshiko Ogata.

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

Communicated by H.-T. Yau

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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

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