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Physics and Mathematics of Quantum Many-Body Systems pp 73–134Cite as

Long-Range Order and Spontaneous Symmetry Breaking in the Antiferromagnetic Heisenberg Model

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Abstract

In the present chapter, which is the main chapter of part I, we develop rigorous and almost complete theories about LRO and SSB in the ground states of the antiferromagnetic Heisenberg model in dimensions two or higher. The problem is subtle and difficult compared with that of the quantum Ising model, because the relevant symmetry (which will be eventually broken) is continuous. But the continuity of the symmetry leads to an interesting object known as the “tower of states”. Section 4.1 is devoted to the discussion about the existence or absence of LRO in the ground state of the antiferromagnetic Heisenberg model. We give a complete proof of the important theorem on the existence of LRO. Then, in Sect. 4.2, we describe in detail the rigorous theory about the “tower of states” and its relation to SSB. In Sect. 4.3, we briefly discuss the mathematical formulation of quantum spin systems in the infinite volume, and see the implications of the results in Sect. 4.2. In the final Sect. 4.4, which is in a sense independent from the rest of the book, we discuss the existence or absence of order in the equilibrium states of the Heisenberg model. We prove two important theorems about the absence of order in two dimensions. One of them, the improved Hohenberg–Mermin–Wagner theorem (Theorem 4.24 in p. 124), is proved here for the first time.

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Notes

  1. 1.

    Note that the choice of the operator \(\hat{A}\) (and hence that of the order operator) is, in a sense, ad hoc. If we set \(\hat{A}=\hat{S}_{\mathrm{tot}}^{(\beta )}\), we then get \(\hat{S}_{\mathrm{tot}}^{(\gamma )}\) as a candidate of the order operator. This is certainly not useful if there is antiferromagnetic ordering, but is a precise choice if one anticipates ferromagnetic ordering.

  2. 2.

    To prove this we note that (2.1.22) implies \((\hat{U}^{(\alpha )}_{\pi /2})^\dagger \,\hat{S}^{(\beta )}_x\,\hat{U}^{(\alpha )}_{\pi /2}=-\sum _{\gamma =1}^3\varepsilon _{\alpha \beta \gamma }\,\hat{S}^{(\gamma )}_x\), and make use of the invariance (2.5.5).

  3. 3.

    We do not prove Theorem 4.2 in the present book. Although the statement is not presented as a mathematical theorem in [58], it can be made rigorous with some effort. See [63] for a more mathematical formulation of Shastry’s argument.

  4. 4.

    Shen, Qiu, and Tian mainly studied the Hubbard model. See Sect. 10.2.3.

  5. 5.

    We do not, however, assume any knowledge about the proof in the rest of the book. The reader may skip the whole proof and jump to Sect. 4.2, where we discuss the important notion of the tower of low-lying states.

  6. 6.

    We here use easily verifiable relations \(L^{-d}\sum _{x\in \varLambda _L}e^{-ik\cdot x}=\delta _{k,o}\) and \(L^{-d}\sum _{k\in {\mathscr {K}}_L}e^{ik\cdot x}=\delta _{x,o}\).

  7. 7.

    That \(I_1=\infty \) suggests that the model with \(d=1\) cannot have long-range order. This is indeed the idea behind Shastry’s theorem, Theorem 4.2.

  8. 8.

    There is a mistake in the calculation in [47], and the authors concluded the existence of order in \(d=2\) only for \(S\ge 3/2\). This mistake was pointed out and corrected in Appendix A of [1]. See also endnote 6 of [39].

  9. 9.

    \(f^{(-1)}_L(k)\) is sometimes called the susceptibility since it represents the second order correction to the ground state energy when the external field \(h\tilde{S}_k\) is applied. We note however that it is not directly related to the physically relevant magnetic susceptibility of an antiferromagnet, since we are here considering the symmetric ground state in a finite volume.

  10. 10.

    Note that the choice \(b_x=(-1)^x b\) is equivalent to \(\lambda =0\), and hence realizes the minimum energy.

  11. 11.

    There indeed is a clever (but difficult) shortcut which almost directly proves (4.1.49) from (4.1.51). See [29]. We present the following longer proof, from which one can learn an idea useful in different applications of the reflection positivity method.

  12. 12.

    It is worth noting that the ferromagnetic Heisenberg model is not reflection positive [12].

  13. 13.

    With some extra effort, one can treat models in which \(\hat{H}^\mathrm {L}_{\varvec{h}}\) or \(\hat{H}^\mathrm {R}_{\varvec{h}}\) are not real. See Appendix 2 of [43].

  14. 14.

    To be more precise, one can make use of three independent low-lying states, namely, \(\hat{{\mathscr {O}}}_L^{(1)}|\varPhi _\mathrm {GS}\rangle \), \(\hat{{\mathscr {O}}}_L^{(2)}|\varPhi _\mathrm {GS}\rangle \), and \(\hat{{\mathscr {O}}}_L^{(3)}|\varPhi _\mathrm {GS}\rangle \). But these are far from enough.

  15. 15.

    More precisely, the states \((\hat{{\mathscr {O}}}_L^+)^M|\varPhi _\mathrm {GS}\rangle \) or \((\hat{{\mathscr {O}}}_L^-)^{|M|}|\varPhi _\mathrm {GS}\rangle \) are nonzero.

  16. 16.

    The existence of the limit is not hard to prove. See, e.g., Appendix of [65].

  17. 17.

    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 of the tower in Table I of [24], and a complete tower structure in Table I of [31], both for the \(S=1/2\) antiferromagnetic Heisenberg model on the square lattice.

  18. 18.

    See, e.g., [9], where the existence of LRO (without SSB) in the antiferromagnetic Heisenberg model on the triangular lattice is investigated numerically.

  19. 19.

    It is unfortunate that our rigorous results in 1994 on Anderson’s tower is not mentioned in this review. We succeeded in proving theorems which had confirmed the standard picture, but were not very successful in conveying the result to the most relevant community.

  20. 20.

    The order of the limits is essential here. If one takes the limit \(k\uparrow \infty \) for finite L one simply gets S, which is the maximum possible value of \(|\hat{{\mathscr {O}}}_L^{(\alpha )}/L^d|\). It does not reflect any properties of the ground state.

  21. 21.

    To be rigorous, \(\lim \) in (4.2.12) and (4.2.13) should be replaced by \(\liminf \).

  22. 22.

    This can be seen by examining the transformation under \(\hat{U}^{\varvec{n}}_\pi =\exp [-i\pi \hat{\varvec{S}}_{\mathrm{tot}}\cdot \varvec{n}]\), the \(\pi \) rotation about the \(\varvec{n}\) axis.

  23. 23.

    One can take arbitrary products of \(\hat{U}^{(\alpha )}_\theta \) with various \(\alpha =1,2,3\) and \(\theta \).

  24. 24.

    See, e.g., Sect. 6.5 (in particular p. 85) of [22].

  25. 25.

    One can construct an artificial model with SU(2) invariance in which (4.2.26) is violated for an obvious reason. See Remark 4 in p. 195 of [34]. This means that general arguments as found in Sect. 4.2.2 are useless; one needs to invoke a strong argument specific to the antiferromagnetic Heisenberg model on the hypercubic lattice.

  26. 26.

    To be rigorous \(\lim \) should be replaced by \(\liminf \).

  27. 27.

    The operator (4.1.4), with \(\varLambda _L\) replaced by \(\varLambda \), is a natural order operator for detecting ferrimagnetic order. It should be noted however that ferrimagnetic order may also be characterized by the ferromagnetic order operator, namely, the total spin.

  28. 28.

    One may say that there are quantum fluctuations in the ground states of antiferromagnetic and ferrimagnetic systems, but not in those of ferromagnetic systems.

  29. 29.

    It is also argued in [54] that the exact ferrimagnetic ground state with the largest eigenvalue of \(\hat{S}^{(3)}_{\mathrm{tot}}\) is a physical (or ergodic) state in which macroscopic quantities exhibit small fluctuations. To prove this conjecture is a challenging problem in mathematical physics.

  30. 30.

    The reader not interested in mathematical details may safely skip this subsection.

  31. 31.

    The existence of the limit \(L\uparrow \infty \), on the other hand, is not guaranteed in general. To be rigorous one should replace \(\lim \) with \(\liminf \). The same comment applies to \(\lim _{L\uparrow \infty }\) that appear in the rest of this section.

  32. 32.

    We should note that, although this general lemma proves the existence of \(M(L)\), it does not tell us whether a concrete choice of \(M(L)\), say \(M(L)=L^2\), is suitable.

  33. 33.

    We learned this method from Hosho Katsura. A similar technique was used by Sannomiya, Katsura, and Nakayama [57]. See (28) and (29) of [57].

  34. 34.

    As in (2.2.5), the operator \(\hat{S}^{(\alpha )}_x\) acts as \(\hat{S}^{(\alpha )}\) on the spin at x and as the identity on other spins.

  35. 35.

    In general a state is said to be translation invariant if \(\rho (\tau _x(\hat{A}))=\rho (\hat{A})\) for any x in \(\mathbb {Z}^d\) rather than in \(\mathbb {Z}^d_{\mathrm{even}}\). We are here making a slight abuse of the terminology.

  36. 36.

    Appendix A of [35] contains a (hopefully accessible) discussion about different definitions of ground states in infinite systems.

  37. 37.

    This terminology has little to do with the ergodic hypothesis, which was once believed to be relevant to the foundation of equilibrium statistical mechanics, but comes from the ergodic theory in mathematics. The ergodic theory of course has its root in Boltzmann’s ergodic hypothesis.

  38. 38.

    See Remark 1 at the end of Sect. 2.5 of [35] for the relation to the standard definition of ergodic states.

  39. 39.

    To be rigorous, this definition depends on the tacit (but essential) assumption that the physical ground states of the model are invariant under the translation by any \(x\in \mathbb {Z}^d_{\mathrm{even}}\), i.e., have period two. There is a logical possibility (which is extremely unlikely but has not been rigorously ruled out yet) that the ground states spontaneously break the translation symmetry and have longer period.

  40. 40.

    Note that this instability does not come from an energetic reason since \(|\varPhi _\mathrm {GS}\rangle \) definitely has the lowest energy. Since \(|\varPhi _\mathrm {GS}\rangle \) may be regarded as a superposition of macroscopically distinct states as in (4.2.21) (see also (3.3.8)), this instability may be closely related to the collapse of Schrödinger’s cat type states. As far as we understand the mechanism of such collapse is still poorly understood. But see [60] for a formulation of the problem and some promising results.

  41. 41.

    The existence of the limit has not been proved. To be rigorous one should replace \(\lim \) by \(\liminf \) or \(\limsup \), or take a suitable subsequence of L. See Theorem A.24 in p. 488 and the discussion that follows.

  42. 42.

    A ground state of a classical spin system is a spin configuration that minimizes the Hamiltonian.

  43. 43.

    The integral which determines \(Z_L^\text {Gauss}(\beta )\) is in fact ill-defined since the integrand \(\exp [-\beta H^\text {Gauss}]\) is invariant under the shift \(\theta _x\rightarrow \theta _x+c\) for all \(x\in \varLambda _L\). But this is a well-known problem, which can be resolved by standard methods. All the calculations can be justified, for example, by using the regularized Hamiltonian \(H^\text {Gauss}=\sum _{\{x,y\}\in {\mathscr {B}}_L}(\theta _x-\theta _y)^2/2+\mu \sum _{x\in \varLambda _L}(\theta _x)^2\) with \(\mu >0\), and letting \(\mu \downarrow 0\) after evaluating the correlations.

  44. 44.

    Let f be a Gaussian random variable such that \(\langle f\rangle =0\) and \(\langle f^2\rangle =a\). It is easily verified that \(\langle f^{2n}\rangle =(2n-1)!!\,a^n\). Then one confirms that \(\langle \cos f\rangle =\sum _{n=0}^\infty (-1)^n\langle f^{2n}\rangle /(2n)!\)

    \(=\sum _{n=0}^\infty (-1)^na^n(2n-1)!!/(2n)!=\sum _{n=0}^\infty (-1)^na^n/(2^n n!)=\exp (-a/2)\).

  45. 45.

    Note first that \(\partial \exp [-\beta H^\text {Gauss}]/\partial \theta _x=\beta \sum _{y\in {\mathscr {N}}(x)}(\theta _y-\theta _x)\exp [-\beta H^\text {Gauss}]\). Denote the integral over all \(\theta _x\) as \(\int d\Theta (\cdots )\). Then by integration by parts we find

    \(\beta \int d\Theta (\theta _o-\theta _z)\sum _{y\in {\mathscr {N}}(x)}(\theta _y-\theta _x)\exp [-\beta H^\text {Gauss}] =\int d\Theta (\theta _o-\theta _z)\partial \exp [-\beta H^\text {Gauss}]/\partial \theta _x =(-\delta _{x,o}+\delta _{x,z})\int d\Theta \exp [-\beta H^\text {Gauss}]\). Dividing by the normalization factor \(Z_L^\text {Gauss}(\beta )=\int d\Theta \exp [-\beta H^\text {Gauss}]\), we get \(\beta \sum _{y\in {\mathscr {N}}(x)}\langle (\theta _o-\theta _z)(\theta _y-\theta _x)\rangle ^\text {Gauss}_\beta =-\delta _{x,o}+\delta _{x,z}\), which is the desired (4.4.16). (To be rigorous this derivation should be first carried out for \(\mu >0\) as in footnote 43.)

  46. 46.

    We have simply used the standard formulas for electric potential generated by a point charge. Note that we do not need to worry about the divergence near the charge since our problem is defined on a lattice. We believe that the estimate can be made rigorous by a suitable asymptotic analysis, but have not worked out the details.

  47. 47.

    Mermin and Wagner clearly state in [45] that the essential idea of the proof was due to Hohenberg. It is unfortunate that the result is often referred to as the Mermin–Wagner theorem.

  48. 48.

    To be rigorous \(\lim _{L\uparrow \infty }\) should be \(\limsup _{L\uparrow \infty }\).

  49. 49.

    As far as we know this improved theorem appears for the first time in the present book. The strongest result in this direction may be that by Fröhlich and Pfister [18], which (roughly) states that any infinite volume equilibrium state of a rotationally invariant two-dimensional model preserves the symmetry.

  50. 50.

    Theorems 4.24 and 4.25 below can be almost automatically extended to a model with the Hamiltonian \(\hat{H}_h=\sum _{\{x,y\}\in {\mathscr {B}}_L}J_{x,y}\,\hat{\varvec{S}}_x\cdot \hat{\varvec{S}}_y-\sum _{x\in \varLambda _L}\varvec{h}_x\cdot \hat{\varvec{S}}_x\), where the interactions \(J_{x,y}\) are arbitrary except that \(|J_{x,y}|\le 1\).

  51. 51.

    It is expected that the quantum XY model, which is obtained by setting \(\lambda =D=0\) in (2.5.14), exhibits a power law decay at sufficiently low temperature. Theorem 4.25 automatically extends to this case.

  52. 52.

    In fact it is enough to count two bonds for general x, but we must count four neighbors of the origin o.

  53. 53.

    To be rigorous one should consider \(\limsup _{L\uparrow \infty }|\langle \hat{S}^{(1)}_o\rangle ^L_{\beta ,h}|\).

  54. 54.

    To be rigorous, \(\lim \) should be replaced by \(\liminf \).

  55. 55.

    It is unfortunate that the method based on reflection positivity is essentially the only method we know to prove the existence of long-range order associated with a breakdown of continuous symmetry.

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    Hal Tasaki

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Tasaki, H. (2020). Long-Range Order and Spontaneous Symmetry Breaking in the Antiferromagnetic Heisenberg Model. 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_4

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