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Affleck–Kennedy–Lieb–Tasaki Model

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Physics and Mathematics of Quantum Many-Body Systems

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Abstract

In 1987, Affleck, Kennedy, Lieb, and Tasaki proposed a one-dimensional \(S=1\) antiferromagnetic quantum spin model, now called the AKLT model, whose ground state can be written down explicitly. It was shown that the energy spectrum of the model has a finite gap above the ground state energy, and the ground state correlation function decays exponentially [4, 5]. These results agree with Haldane’s conclusion for the integer spin antiferromagnetic Heisenberg chain. But note that this does not prove the Haldane “conjecture” since the model is different from the Heisenberg model. Nevertheless this work proves that there exists an \(S=1\) chain that has a unique disordered ground state with a gap, and hence provides a strong support to Haldane’s conclusion. It also provides a starting point of further studies of various phenomena and concepts related to the Haldane gap, as we shall see in Chap. 8. Now it is commonly believed that the AKLT model is (in some sense) at the “center” of the Haldane phase. In Sect. 7.1 we give a detailed discussion of the main results about the AKLT model. In Sect. 7.2 we study some unexpected features of the model, which will turn out to be essential for the study of Haldane phenomena. In Sect. 7.3 we discuss extensions and closely related models.

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Notes

  1. 1.

    From (2.2.19), we see that \(\hat{\varvec{S}}_x\cdot \hat{\varvec{S}}_{x+1}\) takes two eigenvalues, \(-3/4\) and 1/4. The identity then follows.

  2. 2.

    Some exact excited states were obtained in [9, 64, 65]. Unlike in integrable models, it is not possible to obtain general excited states.

  3. 3.

    From numerical works, the energy gap of the AKLT model for large L is estimated to be about 0.7. See, e.g., [32, 34].

  4. 4.

    See footnote 4 in p. 156.

  5. 5.

    In actual magnetic materials, a spin \(S=1\) comes from two \(S=1/2\)’s (of electrons) coupled by Hund’s rule. But this mathematical construction is valid irrespective of the origin of the spin \(S=1\).

  6. 6.

    Proof Because of the SU(2) invariance, it suffices to show that \(\hat{S}^{(3)}{\mathscr {S}}={\mathscr {S}}(\hat{S}^{(3)}_\mathrm{L}+\hat{S}^{(3)}_\mathrm{R})\). This is easily verified by operating the both sides to \(|\psi ^\sigma _\mathrm{L}\rangle |\psi ^{\sigma '}_\mathrm{R}\rangle \).

  7. 7.

    We believe that the term “valence-bond solid state” was first used in [4] to mean the specific class of states, which includes (7.1.12), introduced in [4, 5]. But recently it is also common to call a simple tensor product of valence bonds the VBS state. See footnote 10 below. The state (7.1.12) is also known as the AKLT state.

  8. 8.

    This construction has been implemented in an optical experiment, and the “photonic VBS state” was generated [40]. The experiment is motivated by measurement based quantum computation with the VBS state [17, 60, 63].

  9. 9.

    When L is odd, one can use the state which almost is the Néel state, with one extra 0. The argument is then essentially the same.

  10. 10.

    Here “dimer”, as opposed to “monomer”, stands for a pair of neighboring sites. The product state of singlet pairs which cover the whole lattice site is called the dimer state (or, more precisely, the dimer covering state). It is sometimes called the VBS (valence-bond solid) state. See footnote 7 above.

  11. 11.

    This should be obvious. To be rigorous, we are using Lemma A.10 in p. 469.

  12. 12.

    This is guaranteed since the ring of polynomials (with complex coefficients) is in a unique factorization domain [89].

  13. 13.

    See [7, 35, 47,48,49,50] for refinements and extensions of Knabe’s method, e.g., to systems with open boundaries or in higher dimensions. A related but different method for proving the existence of a finite gap in higher dimensional models is employed in [1, 71, 72]. See Sect. 7.3.2.

  14. 14.

    The ground states are degenerate. See Sect. 7.2.3.

  15. 15.

    Exactly as in Problem 2.5.c (p. 39), one can show that \(\langle \varPhi _\mathrm{VBS}|(\hat{S}^{(3)}_x)^2|\varPhi _\mathrm{VBS}\rangle =2/3\). This means that the average densities of \(+\), −, and 0 are all equal to 1/3. See also Problem 7.2.2.c (p. 200).

  16. 16.

    The term “quantum spin liquid” is used to indicate a variety of spin systems which exhibit liquid-like behavior. See, e.g., the review articles [59, 76].

  17. 17.

    This corresponds to the definition (3.4.6) of long-range order, which is expected to be equivalent to the definition (3.4.3) used throughout Part I. To be rigorous the existence of the limits in (7.2.4) and (7.2.7) are note proved in general.

  18. 18.

    Here the existence of the limits in (7.2.7) can be proved.

  19. 19.

    The reader anxious to study another exotic property of the AKLT model may skip this subsection and jump to Sect. 7.2.3 (and come back here when necessary).

  20. 20.

    The structure essentially equivalent to matrix product states was already known in the context of classical statistical mechanics. The work by Baxter [13] is an early example.

  21. 21.

    In the original works [26,27,28], matrix product states were called finitely correlated states. The term matrix product states was introduced later in [43]. See also [82, 83], where the representation was rediscovered and put into a slightly different context.

  22. 22.

    The general construction of matrix product states given in [28] has two parents: the AKLT model and Accardi’s proposal for the construction of quantum Markov chains [2, 3]. The relationship between the two is clarified in [66]. Note, however, that there are several definitions of quantum Markov chains that continue to be used in parallel. For example, in the quantum information community a common definition is given in terms of the case of equality in the strong subadditivity property of entropy [38].

  23. 23.

    The derivation is by no means the unique or the most convenient way to derive the matrix product representation of the VBS state. See Sect. 4.1.5 of [77] for a simple derivation which starts from the expression (7.1.11) of the pre-VBS state (which is (84) of [77]), and [80] for a derivation based on the Schwinger boson representation of [10].

  24. 24.

    The spin configuration \(\varvec{\sigma }=(0,\ldots ,0)\) is the only exception. It is generated both by \((1,\ldots ,1)\) and \((2,\ldots ,2)\).

  25. 25.

    These states satisfy \(\hat{S}^{(\alpha )}|p^{(\alpha )}\rangle =0\) and \(\exp [-i(\pi /2)\hat{S}^{(\alpha )}]\,|p^{(\beta )}\rangle =\sum _{\gamma =1,2,3}\varepsilon _{\alpha \beta \gamma }\,|p^{(\gamma )}\rangle \) for any \(\alpha ,\beta ,\gamma =1,2,3\), where \(\varepsilon _{\alpha \beta \gamma }\) is the Levi-Civita symbol (see (A.3.1)).

  26. 26.

    It is natural to represent (in general) \((A^\sigma _{\alpha ,\alpha '})^*\) as

  27. 27.

    In (7.2.27), (7.2.28), and (7.2.29), we used a notation that is valid when the operator \(\hat{S}^{(3)}\) is replaced by an arbitrary local operator. The double sum in the second expression of (7.2.29) is not necessary if we are interested only in \(\hat{S}^{(3)}\).

  28. 28.

    Suppose we start from the vector \((0,1,0)^\mathrm{t}\). Multiplying it by \(\mathsf {A}^\downarrow \), we get \((0,0,1)^\mathrm{t}\). Since further multiplication by \(\mathsf {A}^\downarrow \) only yields zero, we multiply the result by \(\mathsf {A}^\uparrow \) to get \((0,1/\sqrt{2},0)^\mathrm{t}\). Note that the vector is now proportional to the initial one. Going back to the initial vector \((0,1,0)^\mathrm{t}\), we can also multiply it by \(\mathsf {A}^\uparrow \) to get \((-1,0,0)^\mathrm{t}\), and then by \(\mathsf {A}^\downarrow \) to get \((0,-1/\sqrt{2},0)^\mathrm{t}\). In this way we have formed a singlet pair \(\{|\!\uparrow \rangle |\!\downarrow \rangle -|\!\downarrow \rangle |\!\uparrow \rangle \}/\sqrt{2}\).

  29. 29.

    See [28, 70]. For mathematical details see also Sect. 6.3 of [90].

  30. 30.

    The eigenvector corresponding to \(\lambda \) is \(\delta _{\alpha ',\beta '}\) since \(\sum _{\alpha ',\beta '}(\tilde{\mathsf {A}})_{\alpha ,\beta ;\alpha ',\beta '}\,\delta _{\alpha ',\beta '} =\sum _{\alpha ',\sigma }(A^\sigma _{\alpha ,\alpha '})^*\,A^\sigma _{\beta ,\alpha '} =\sum _\sigma (\mathsf {A}^\sigma (\mathsf {A}^\sigma )^\dagger )_{\beta ,\alpha }=\lambda \,\delta _{\alpha ,\beta }\).

  31. 31.

    The condition (7.2.41) is valid with \(\lambda =3/4\). The four matrices \(\mathsf {A}^0\mathsf {A}^+\), \(\mathsf {A}^0\mathsf {A}^-\), \(\mathsf {A}^+\mathsf {A}^-\), and \(\mathsf {A}^-\mathsf {A}^+\) are enough to span the whole \(2\times 2\) matrices; the condition (i) is verified with \(\ell _0=2\).

  32. 32.

    The condition (7.2.41) is valid with \(\lambda =1\). But we find, with the aid of a computer, that the transfer matrix \(\tilde{\mathsf {A}}\) has eigenvalues \(\pm 1\).

  33. 33.

    To be precise, let \(\rho [\cdot ]\) be an arbitrary translation invariant state on the infinite chain. (See Appendix A.7 for the notation.) Then one can take a sequence \(\rho _j[\cdot ]\) with \(j=1,2,\ldots \) of matrix product states on the infinite chain such that \(\lim _{j\uparrow \infty }\rho _j[\hat{A}]=\rho [\hat{A}]\) for each \(\hat{A}\in \mathfrak {A}_\mathrm{loc}\).

  34. 34.

    A general construction (which may not give the simplest parent Hamiltonian) is as follows. Take the open chain \(\{x,\ldots ,x+\ell _0\}\) with \(\ell _0+1\) sites, where \(\ell _0\) is the constant determined in Theorem 7.5, and consider the corresponding matrix product states (7.2.45) with all possible boundary terms. Let \(\hat{P}_x\) be the projection onto the space spanned by all these matrix product states. Then \(\hat{H}=-\sum _x\hat{P}_x\) gives a desired parent Hamiltonian.

  35. 35.

    See footnote 4 in p. 156.

  36. 36.

    It may be useful to know that our \(\tilde{\mathsf {A}}\) and \(\tilde{\mathsf {B}}\) become \(\mathbb {E}_\mathbbm {1}\) and \(\mathbb {E}_{\hat{S}^{(3)}}\), respectively, in [28].

  37. 37.

    We learned this from Hosho Katsura. See [48] for recent refinements of the method.

  38. 38.

    Examine the representation (7.2.1), and see what type of spin configurations one gets for each state.

  39. 39.

    One can of course write down a formula corresponding to (7.3.1), exactly as we did in (7.1.12) for the \(S=1\) chain.

  40. 40.

    More generally (7.3.2) is expressed in terms of a polynomial including \((\hat{\varvec{S}}_x\cdot \hat{\varvec{S}}_{x+1})^4\).

  41. 41.

    For simplicity we only discuss VBS states were each bond of the lattice carries one valence bond. Clearly one can define more general VBS states with multiple valence bonds on each lattice bond as in (7.3.1).

  42. 42.

    We assume that the boundary conditions are chosen so that the lattice is bipartite.

  43. 43.

    Note that our uniqueness theorem is limited to translation invariant states. We still do not have a stronger uniqueness theorem comparable to Theorem 7.2 in p. 179.

  44. 44.

    It was demonstrated in [12, 21, 22] that whether a given quantum spin system has a gap or not (in the infinite volume) can be an undecidable problem. Although these results are highly nontrivial and deep, the difficulty related to the gap in the hexagonal lattice AKLT model has nothing to do with the undecidability. (And the existence of a gap was finally (almost) decided. See below.) 

  45. 45.

    First part of the work is fully rigorous, but the second part is numerical. Therefore the lower bound for the energy gap, although being reliable and conclusive, is not yet mathematically rigorous.

  46. 46.

    Again the first part is rigorous and the second part is numerical. But this work is closer to (or may be regarded as) a computer-aided proof since only the Lanczos method is used in the numerical part.

  47. 47.

    One might wonder if the two-dimensional VBS state exhibits exotic properties similar to those of its one-dimensional counterpart. It is obvious from the construction that the state defined on a finite lattice with open boundary is accompanied by edge spins that live on boundary sites. On the other hand it is likely that the state does not have a simple hidden order as in the one-dimensional VBS state. This observation is consistent with the modern picture that the two dimensional VBS state has only weak SPT (symmetry protected topological) order. See the end of Sect. 8.3.4.

  48. 48.

    If all sites are identical the tensor is simply \(A^{\sigma }_{(\alpha _{x,y})_{y\in {\mathscr {N}}(x)}}\).

  49. 49.

    See [11] for experimental realizations of the Briegel–Raussendorf state.

  50. 50.

    In the original paper by Briegel and Raussendorf [19], the term “cluster state” was introduced primarily to indicate the state (7.3.29) defined on a connected percolation cluster. See also [18].

  51. 51.

    In Sects. 8.2.3 and 8.3.2, in connection with the notion of symmetry protected topological (SPT) phase, we argue that the invariance under the \(\mathbb {Z}_2\times \mathbb {Z}_2\) transformation is an essential property of the VBS state and the AKLT model. The state (7.3.16) and the corresponding Hamiltonian (7.3.21) also have \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry. More precisely, when L is even, the Hamiltonian (7.3.21) commutes with both \(\hat{U}_\mathrm{odd}:=\prod _{\ell =1}^{L/2}\hat{\sigma }^{(1)}_{2\ell -1}\) and \(\hat{U}_\mathrm{even}:=\prod _{\ell =1}^{L/2}\hat{\sigma }^{(1)}_{2\ell }\). For the aspect of the Briegel–Raussendorf as a state in a nontrivial symmetry protected topological phase, see, e.g., [78].

  52. 52.

    Let us start from the vector \((1,0)^\mathrm{t}\). The vector does not change when we multiply it (repeatedly) by \(\mathsf {A}^\uparrow \). When we multiply \((1,0)^\mathrm{t}\) by \(\mathsf {A}^\downarrow \), it changes to \((0,1)^\mathrm{t}\), passing information to the left that there is a \(\downarrow \). A multiplication of \((0,1)^\mathrm{t}\) by \(\mathsf {A}^\uparrow \) simply reverts it to \((1,0)^\mathrm{t}\). But when we multiply \((0,1)^\mathrm{t}\) by \(\mathsf {A}^\downarrow \), we get \((0,-1)^\mathrm{t}\) which contains the desired phase factor (and still the information that there is a \(\downarrow \) is passed to the left).

  53. 53.

    Note that our usage of the term cluster is very much different from that in the original paper by Briegel and Raussendorf [19].

  54. 54.

    We find that the names “graph state” and “hyper graph state” are too general and do not characterize the particular states (7.3.29) or (7.3.33) very well. Since these states are characterized by the quantum mechanical phase associated with each cluster, we feel that the name “cluster-phase state” is somewhat more appropriate. In this terminology one may specify the type of cluster by talking about the bond-phase state, the triangle-phase state, or the plaquette-phase state. The  bond-phase state is of course the Briegel–Raussendorf state (7.3.29). (It is unfortunate that the term “cluster phase” (without hyphen) is used, e.g., in [74] to mean a symmetry protected topological phase that includes the cluster state.)

  55. 55.

    The state is called the triangle-phase state in the terminology proposed in footnote 54.

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    Hal Tasaki

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Tasaki, H. (2020). Affleck–Kennedy–Lieb–Tasaki 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_7

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