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Ground States of the Antiferromagnetic Heisenberg Chains

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Physics and Mathematics of Quantum Many-Body Systems

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Abstract

In the present chapter, we focus on properties of the ground state and low energy excited states of the one-dimensional antiferromagnetic Heisenberg model, often called the “antiferromagnetic Heisenberg chain”. Recall that, according to Shastry’s theorem (Theorem 4.2 in p. 76), the ground state does not exhibit long-range order or spontaneous symmetry breaking. In Sect. 6.1, we describe Haldane’s conclusion about the qualitative difference between the antiferromagnetic Heisenberg chains with half-odd-integer and integer S, along with necessary background. Then we discuss the origin of the difference from two different points of view. We describe and prove the (generalized) Lieb–Schultz–Mattis theorem in Sect. 6.2, and present a semi-classical picture based on kink dynamics in Sect. 6.3.

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Notes

  1. 1.

    Note that the numbering of lattice sites is different from that in (3.1.2).

  2. 2.

    Such results based on unproved assumptions are sometimes referred to as “exact but non-rigorous” results.

  3. 3.

    There is a set of energy eigenstates, known as the des Cloizeaux–Pearson mode, with excitation energy \(\varepsilon _k=(\pi /2)|\sin k|\), where k is the wave number [17].

  4. 4.

    There is no general proof of this fact. In the quantum Ising model (3.3.1) with sufficiently large \(\lambda \), the Ornstein–Zernike type behavior of the ground state correlation function was rigorously established by Kennedy [34].

  5. 5.

    See footnote 5 in p. 53.

  6. 6.

    We learned that Edward Witten, who heard Haldane’s talk in 1983 at the conference in honor of the 60th birthday of Philip Anderson, was one of the first to realize a possible connection between Haldane’s argument and the topological angle. Affleck was informed of Haldane’s work and its possible topological interpretation by Witten (Ian Affleck, private communication).

  7. 7.

    It is indeed surprising that Haldane’s two published papers [25, 26] deal with large S limit, where the difference between the models with half-odd-integer S and integer S is expected to become infinitesimal, as is suggested by the formula \({\varDelta E}\simeq 2Se^{-\pi S}\) for the Haldane gap for integer S. Now it is known that Haldane had yet another argument which relies on the analogy to the Berezinskii–Kosterlitz–Thouless transition, and does not make use of the large S limit. This work has not been published in a journal but was made public by Haldane himself in 2016 [24]. (According to Haldane one of the referees of the paper stated that it was “in manifest contradiction to fundamental principles of physics” [28].) See also [29].

  8. 8.

    It is believed that a three dimensional spin system obtained by weakly coupling the \(S=1\) antiferromagnetic Heisenberg chains has a disordered ground state with a gap [3, 38, 52]. It is likely that NENP never develops long-range order at any temperatures.

  9. 9.

    One finds \({\varDelta E}_\mathrm {H}\simeq 0.089\) for \(S=2\), \({\varDelta E}_\mathrm {H}\simeq 0.0100\) for \(S=3\), and \({\varDelta E}_\mathrm {H}\simeq 7.99\times 10^{-4}\) for \(S=4\). The result for \(S=4\) was obtained by a Monte Carlo simulation of the chain with 73,728 sites, performed on a supercomputer [62].

  10. 10.

    Use the variational estimate described in Sect. .

  11. 11.

    We learned this trick from [36]. In fact this step can be avoided, and one can bound the expectation value \(\langle \varPhi _\mathrm {GS}|(\hat{U}_\mathrm{LSM}^\dagger \hat{H}\hat{U}_\mathrm{LSM}-\hat{H})|\varPhi _\mathrm {GS}\rangle \) directly by using a slightly complicated estimate that makes use of an extra symmetry of \(|\varPhi _\mathrm {GS}\rangle \) as in [5, 39]. We then get a better bound \(\langle \varPhi _\mathrm {LSM}|\hat{H}|\varPhi _\mathrm {LSM}\rangle -E_\mathrm{GS}\le 4\pi ^2S^2/L\) instead of (6.2.11).

    In all the early applications of the Lieb–Schultz–Mattis method [5, 39, 45, 69], it was assumed that the system possesses inversion or time-reversal symmetry in addition to the U(1) symmetry and translation invariance. The above estimate shows that the inversion or time-reversal symmetry is in fact not necessary. It is interesting that this (more or less standard) argument had been overlooked until Koma [36] applied the Lieb–Schultz–Mattis method to quantum Hall systems on a quasi one-dimensional strip.

  12. 12.

    The proof is standard (and essentially the same as that described in footnote 9 in p. 58). Translation invariance \(\hat{T}\hat{H}\hat{T}^\dagger =\hat{H}\) implies \(\hat{T}\hat{H}=\hat{H}\hat{T}\). Thus \(\hat{H}(\hat{T}|\varPhi _\mathrm {GS}\rangle )=\hat{T}\hat{H}|\varPhi _\mathrm {GS}\rangle =E_\mathrm{GS}\hat{T}|\varPhi _\mathrm {GS}\rangle \), and the uniqueness implies \(\hat{T}|\varPhi _\mathrm {GS}\rangle =e^{i\alpha }|\varPhi _\mathrm {GS}\rangle \). The Marshall–Lieb–Mattis theorem (Theorem 2.2 in p. 39) implies \(e^{i\alpha }=\pm 1\), but we do not use this fact.

  13. 13.

    The same statement was proved for the infinite system by Affleck and Lieb [5]. See the end of the present section.

  14. 14.

    Such a statement should be proved for the infinite system. The proof is difficult since one cannot exclude the possibility that linear combinations of the ground state and low-lying excitations for finite L converge to multiple infinite volume ground states in the limit \(L\uparrow \infty \).

  15. 15.

    See Sect. A.2.3 for the meaning of inequalities between self-adjoint operators.

  16. 16.

    It is believed that \(\langle \varPhi _\mathrm {GS}|\varPhi _\mathrm{LSM}\rangle \simeq -1\) if the ground state exhibits Haldane gap [42].

  17. 17.

    Equation (6.2.29) is easily verified by differentiating both sides by \(\varDelta \theta \).

  18. 18.

    This rephrasing of the Lieb–Schultz–Mattis theorem was also essential for the later development of related theories. See the next paragraph and Sect. 8.3.5.

  19. 19.

    One may call a bond right to an even site even.

  20. 20.

    We admit this argument is far from sufficient to conclude that the model is gapless.

  21. 21.

    We note that this may not be the case in general. In this sense our picture here for the \(S=1\) chain is only approximate.

  22. 22.

    This proposition was proved by the present author in 1986, but was not published. It was later included as a part of [58].

  23. 23.

    The proof is elementary. Let \(|\varPsi _j\rangle \) with \(j=1,2,\ldots ,D\) be the normalized energy eigenstate with eigenvalue \(E_j\). We choose \(|\varPsi _1\rangle =|\varPhi _\mathrm {GS}\rangle \). Then \(e^{-\beta \hat{H}}|\tilde{\varPsi }^{\varvec{\sigma }^{(0)}}\rangle =\sum _{j=1}^D|\varPsi _j\rangle e^{-\beta E_j}\langle \varPsi _j|\tilde{\varPsi }^{\varvec{\sigma }^{(0)}}\rangle =e^{-\beta E_\mathrm{GS}}\{|\varPhi _\mathrm {GS}\rangle \langle \varPhi _\mathrm{GS}|\tilde{\varPsi }^{\varvec{\sigma }^{(0)}}\rangle +\sum _{j=2}^D e^{-\beta (E_j-E_\mathrm{GS})}|\varPsi _j\rangle \langle \varPsi _j|\tilde{\varPsi }^{\varvec{\sigma }^{(0)}}\rangle \}\). Noting that \(e^{-\beta (E_j-E_\mathrm{GS})}\downarrow 0\) as \(\beta \uparrow \infty \) if \(j\ge 2\), we get (6.3.10).

  24. 24.

    There are many ways to arrange the product in the right-hand side. One chooses an arrangement suitable for the purpose. See, e.g., [21, 35, 56, 58].

  25. 25.

    In fact it is enough to sum over \(\varvec{\sigma }^{(j)}\) such that \(\overline{\varvec{\sigma }^{(j)}}=0\).

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Tasaki, H. (2020). Ground States of the Antiferromagnetic Heisenberg Chains. In: Physics and Mathematics of Quantum Many-Body Systems. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-41265-4_6

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