Skip to main content
SpringerLink
Log in

Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system with global U(1) symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the “filling factor” is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb–Schultz–Mattis and related theorems.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. As far as we know this implication of the local version of the Lieb–Schultz–Mattis theorem has not been pointed out before. A closely related observation for infinite systems was made by Koma [5].

  2. In [4] the hopping amplitude was assumed to be real to ensure time-reversal symmetry. This condition was removed in [5].

  3. To be very precise, this quantity may be slightly different from the variance since \(\nu \) may differ from \(\left\langle \Phi _L^\mathrm{GS}\left| \hat{\rho }_\ell \right| \Phi _L^\mathrm{GS}\right\rangle \) by O(1 / L) as in (6). But this difference is negligible for large L.

  4. To be precise, one lets \({\Delta }\hat{\rho }_\ell ':=\hat{\rho }_\ell -\left\langle \Phi _L^\mathrm{GS}\left| \hat{\rho }_\ell \right| \Phi _L^\mathrm{GS}\right\rangle ={\Delta }\hat{\rho }_\ell +O(1/L)\), and considers the trial state \(|\Xi \rangle ={\Delta }\hat{\rho }_\ell '{|{\Phi _L^{\mathrm {GS}}}\rangle }/\sqrt{\left\langle \Phi _L^\mathrm{GS}\left| ({\Delta }\hat{\rho }_\ell ')^2\right| \Phi _L^\mathrm{GS}\right\rangle }\), which obviously satisfies \(\langle \Phi ^\mathrm{GS}_L|\Xi \rangle =0\), and also satisfies \(\langle \Gamma |\hat{H}_L|\Gamma \rangle -E_L^\mathrm{GS}\le (\text {const})\times \ell ^{-2}\). To show the latter variational estimate, we use the identity \(\langle \Xi |\hat{H}_L|\Xi \rangle -E_L^\mathrm{GS}=(1/2)\left\langle \Phi _L^\mathrm{GS}\left| [[{\Delta }\hat{\rho }_\ell ',\hat{H}_L],{\Delta }\hat{\rho }_\ell ']\right| \Phi _L^\mathrm{GS}\right\rangle /\left\langle \Phi _L^\mathrm{GS}\left| ({\Delta }\hat{\rho }_\ell ')^2\right| \Phi _L^\mathrm{GS}\right\rangle \), and note that \(\Vert [[{\Delta }\hat{\rho }_\ell ',\hat{H}_L],{\Delta }\hat{\rho }_\ell ']\Vert =O(\ell ^{-2})\).

  5. Of course \({|{\Phi _L^{\mathrm {GS}}}\rangle }\) and \(\hat{W}_\ell {|{\Phi _L^{\mathrm {GS}}}\rangle }\) may differ globally even when \(\hat{W}_\ell \) is local. See example 2 in Sect. 2.3.

  6. More precisely we mean that the number of near ground states are bounded when L grows.

  7. Antiferromagnetic quantum spin chain with \(S=1\) exhibiting Haldane phenomena is known to possess a kind of hidden order. But this is related to symmetry protected topological phase, not to topological order. See, e.g., [16, 17].

  8. If one adds a small hopping \(t_{j,j+1}=t\ne 0\), then one of \(|\Phi _L^\mathrm{GS,\pm }\rangle \) becomes the unique ground state and the other becomes the near ground state with almost degenerate energy.

  9. This seemingly trivial observation in [5] was essential in removing the assumption about time-reversal (or inversion) symmetry.

  10. For any operators \(\hat{A}\), \(\hat{B}\), one has \(\bigl |\langle \hat{A}^\dagger \hat{B}\rangle \bigr |^2\le \langle \hat{A}^\dagger \hat{A}\rangle \langle \hat{B}^\dagger \hat{B}\rangle \), where \(\langle \cdots \rangle \) is any expectation value.

  11. Note that this interaction can be written as \(\varvec{D}_{j,k}\cdot (\hat{\varvec{S}}_j\times \hat{\varvec{S}}_k)\) with \(\varvec{D}_{j,k}=(0,0,\tilde{J}_{j,k})\). Such an interaction which is not invariant under inversion \((j,k)\rightarrow (k,j)\) was not treated in the earlier works [1,2,3,4]. The extension was made possible by the work of Koma [5]. See also [6]. It is also possible to include the scalar chirality term \(J'_{j,k,\ell }\hat{\varvec{S}}_j\cdot (\hat{\varvec{S}}_k\times \hat{\varvec{S}}_\ell )\).

  12. For a large class of antiferromagnetic chains with vanishing magnetic field, including the model (47) with \(H=0\) and even L, one can show that the ground state is unique and belongs to the sector with \(M=0\). See [1, 23], Sect. 2.1 of [2], and Remark 2 in Section 3.1 of [24].

  13. In the variational estimate as in Lemma 1, it is easier to use the representation of the Hamiltonian (46) in terms of the \(\hat{S}^\pm _j\) operators, and use the relation corresponding to (17).

References

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. 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 

  4. 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 

  5. Koma, T.: Spectral gaps of quantum Hall systems with interactions. J. Stat. Phys. 99, 313–381 (2000). arXiv:cond-mat/9809228

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Nomura, K., Morishige, J., Isoyama, T.: Extension of the Lieb-Schultz-Mattis theorem. J. Phys. A. 48, 375001 (2015). arxiv:1503.05662

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

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

    Article  ADS  Google Scholar 

  9. 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 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. 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 

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

  13. Tasaki, H.: Low-lying excitations in one-dimensional lattice electron systems. Preprint. arXiv:cond-mat/0407616 (2004)

  14. 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 (1983)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  16. Wen, X.-G: Zoo of quantum-topological phases of matter. Preprint. arxiv.org/abs/1610.03911 (2016)

  17. 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.org/abs/1508.02595

  18. Koma, T., Tasaki, H.: Symmetry breaking and finite-size effects in quantum many-body systems. J. Stat. Phys. 76, 745–803 (1994). arXiv:cond-mat/9708132

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Watanabe, H.: Energy gap of neutral excitations implies vanishing charge susceptibility. Phys. Rev. Lett. 118, 117205 (2017). arXiv:1609.09543

    Article  ADS  Google Scholar 

  20. Wen, X.G.: Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons. Oxford University Press, Oxford (2007)

    Book  Google Scholar 

  21. 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 

  22. Oshikawa, M.: Private communication (1997)

  23. Lieb, E.H., Mattis, D.: Ordering energy levels in interacting spin chains. J. Math. Phys. 3, 749–751 (1962)

    Article  ADS  MATH  Google Scholar 

  24. Kennedy, T., Tasaki, H.: Hidden symmetry breaking and the Haldane phase in \(S = 1\) quantum spin chains. Comm. Math. Phys. 147, 431–484 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I wish to thank Tohru Koma, Masaki Oshikawa, and Haruki Watanabe for valuable discussions which were essential for the present work, and Ian Affleck, Hosho Katsura, Tomonari Mizoguchi, Lee SungBin, Akinori Tanaka, Masafumi Udagawa, and Masanori Yamanaka for useful discussions and correspondences. The present work was supported by JSPS Grants-in-Aid for Scientific Research No. 16H02211.

Author information

Authors and Affiliations

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

    Hal Tasaki

Authors
  1. Hal Tasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hal Tasaki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tasaki, H. Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems. J Stat Phys 170, 653–671 (2018). https://doi.org/10.1007/s10955-017-1946-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-017-1946-0

Search

Navigation