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Long-Range Order, “Tower” of States, and Symmetry Breaking in Lattice Quantum Systems

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Abstract

In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits long-range order (LRO) but does not show spontaneous symmetry breaking (SSB). Typical examples include antiferromagnetic quantum spin systems with Néel order, and lattice boson systems which exhibits Bose–Einstein condensation. By extending and improving previous results by Horsch and von der Linden and by Koma and Tasaki, we here develop a fully rigorous and almost complete theory about the relation between LRO and SSB in the ground state of a finite system with continuous symmetry. We show that a ground state with LRO but without SSB is inevitably accompanied by a series of energy eigenstates, known as the “tower” of states, which have extremely low excitation energies. More importantly, we also prove that one gets a physically realistic “ground state” by taking a superposition of these low energy excited states.

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Notes

  1. These energy eigenstates must not be confused with the spin-wave excitations. See the remark at the end of Sect. 3.3.

  2. Here we mean the standard textbook treatment of superconductivity where dynamical electromagnetic field is not included.

  3. It seems that people started observing the tower structure numerically in the early 90s when sufficiently advanced computers became available. We find, for example, partial data for the tower in Table I of [8], and a complete tower structure in Table I of [9], both for the \(S=1/2\) antiferromagnetic Heisenberg model on the square lattice.

  4. The quotation marks indicated that they are not ground states in the standard definition in quantum mechanics.

  5. The commutation relation \([\hat{\mathcal{O}}_L^{(1)},\hat{\mathcal{O}}_L^{(2)}]=i\hat{C}_L\) makes our discussion considerably simple, but may not be necessary. We expect that one can prove basically the same results (with much more effort) without assuming it.

  6. This condition about the support of \(\hat{o}^{(1)}_x\) is introduced to make the proofs, especially that of Lemmas 4.4 and 4.5, easy. One can extend the theory to cover \(\hat{o}^{(1)}_x\) acting on more than one site, by constructing slightly more complicated inductive proof.

  7. The constants depend only on \(o_0\), \(h_0\), \(\zeta \), and \(q_0\). See the proof, for example (4.56), for explicit dependences.

  8. The condition \(M_L=0\) can be replaced by the condition that \(|M_L|\) is bounded by a constant independent of L.

  9. To be more precise it is crucial that the excitation is spread almost uniformly over the whole lattice in the state \(|\Gamma _L^M\rangle \). A state obtained by, for example, exciting a single spin from the ground state satisfies a similar bound as (3.9), but it is regarded as an excited state.

  10. The existence of the limit \(L\uparrow \infty \), on the other hand, is not guaranteed in general (although it is very much expected). One may take a subsequence or replace \(\lim \) with \(\limsup \) or \(\liminf \) if necessary.

  11. The order of the limits is essential here. If one takes the limit \(k\uparrow \infty \) for finite L one simply gets the maximum possible value S, which does not reflect the properties of the ground state.

  12. The argument was used in the proof of Theorem 1.3 of [1] and in Sect. 6 of [15]. We here present a refined version.

  13. We learned this method from Hosho Katsura. A similar technique was used by Sannomiya et al. [26]. See (28) and (29) of [26].

References

  1. Dyson, F.J., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–382 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  2. Neves, E.J., Perez, J.F.: Long range order in the ground state of two-dimensional antiferromagnets. Phys. Lett. 114A, 331–333 (1986)

    Article  ADS  Google Scholar 

  3. Mazurenko, A., Chiu, C.S., Ji, G., Parsons, M.F., Kanász-Nagy, M., Schmidt, R., Grusdt, F., Demler, E., Greif, D., Greiner, M.: A cold-atom Fermi-Hubbard antiferromagnet. Nature 545, 462–466 (2017). arXiv:1612.08436

    Article  ADS  Google Scholar 

  4. Marshall, W.: Antiferromagnetism. Proc. R. Soc. A 232, 48 (1955)

    ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  6. Anderson, P.W.: An approximate quantum theory of the antiferromagnetic ground state. Phys. Rev. 86, 694 (1952)

    Article  ADS  MATH  Google Scholar 

  7. Anderson, P.W.: Basic Notions of Condensed Matter Physics. Westview Press/Addison-Wesley, Boulder (1997)

    Google Scholar 

  8. Gross, M., Sánchez-Velasco, E., Siggia, E.D.: Spin-wave velocity and susceptibility for the two-dimensional Heisenberg antiferromagnet. Phys. Rev. B 40, 11328–1130 (1989)

    Article  ADS  Google Scholar 

  9. Kikuchi, M., Okabe, Y., Miyashita, S.: On the ground-state phase transition of the spin 1/2 XXZ model on the square lattice. J. Phys. Soc. Jpn. 59, 492 (1990)

    Article  ADS  Google Scholar 

  10. Bernu, B., Lhuillier, C., Pierre, L.: Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices. Phys. Rev. Lett. 69, 2590–2593 (1992)

    Article  ADS  Google Scholar 

  11. Azaria, P., Delamotte, B., Mouhanna, D.: Spontaneous symmetry breaking in quantum frustrated antiferromagnets. Phys. Rev. Lett. 70, 2483–2486 (1993)

    Article  ADS  Google Scholar 

  12. Lhuillier, C.: Frustrated Quantum Magnets. Lecture Notes at “Ecole de troisieme cycle de Suisse Romande”. arXiv:cond-mat/0502464 (2002)

  13. Horsch, P., von der Linden, W.: Spin-correlations and low lying excited states of the spin-1/2 Heisenberg antiferromagnet on a square lattice. Z. Phys. B 72, 181–193 (1988)

    Article  ADS  Google Scholar 

  14. Kaplan, T.A., Horsch, P., von der Linden, W.: Order parameter in quantum antiferromagnets. J. Phys. Soc. Jpn. 11, 3894–3898 (1989)

    Article  ADS  Google Scholar 

  15. Koma, T., Tasaki, H.: Symmetry breaking in Heisenberg antiferromagnets. Commun. Math. Phys. 158, 191–214 (1993). https://projecteuclid.org/euclid.cmp/1104254136

  16. 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 

  17. Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008). arXiv:0704.3011

    Article  ADS  Google Scholar 

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

  19. Mattis, D.: Ground-state symmetry in XY model of magnetism. Phys. Rev. Lett. 42, 1503 (1979)

    Article  ADS  Google Scholar 

  20. Nishimori, H.: Spin quantum number in the ground state of the Mattis-Heisenberg model. J. Stat. Phys. 26, 839–845 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  21. Kennedy, T., Lieb, E.H., Shastry, B.S.: Existence of Néel order in some spin-\(1/2\) Heisenberg antiferromagnets. J. Stat. Phys. 53, 1019 (1988)

    Article  ADS  Google Scholar 

  22. Kennedy, T., Lieb, E.H., Shastry, B.S.: The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61, 2582 (1988)

    Article  ADS  Google Scholar 

  23. Kubo, K., Kishi, T.: Existence of long-range order in the XXZ model. Phys. Rev. Lett. 61, 2585 (1988)

    Article  ADS  Google Scholar 

  24. Ozeki, Y., Nishimori, H., Tomita, Y.: Long-range order in antiferromagnetic quantum spin systems. J. Phys. Soc. Jpn. 58, 82–90 (1989). https://doi.org/10.1143/JPSJ.58.82

    Article  ADS  Google Scholar 

  25. Koma, T.: Maximum spontaneous magnetization and Nambu-Goldstone mode. arXiv:1712.09018 (2018) (preprint)

  26. Sannomiya, N., Katsura, H., Nakayama, Y.: Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion. Phys. Rev. D 95, 065001 (2017). arXiv:1612.02285

    Article  ADS  Google Scholar 

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Acknowledgements

It is a pleasure to thank Hosho Katsura for his essential contribution in the derivation of (4.43), which considerably simplified the proof, and for useful comments. I also thank Tohru Koma and Haruki Watanabe for indispensable discussions and comments which made the present work possible, and Masaki Oshikawa and Masafumi Udagawa for useful discussions. The present work was supported by JSPS Grants-in-Aid for Scientific Research no. 16H02211.

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

    Hal Tasaki

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Tasaki, H. Long-Range Order, “Tower” of States, and Symmetry Breaking in Lattice Quantum Systems. J Stat Phys 174, 735–761 (2019). https://doi.org/10.1007/s10955-018-2193-8

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