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  • Dimerization and Néel order in different quantum spin chains through a shared loop representation

    Michael Aizenman, Hugo Duminil-Copin, Simone Warzel∗

    Feb. 5, 2020

    Abstract: The ground states of the spin-S antiferromagnetic chain HAF with a projection-based interaction and the spin-1/2 XXZ-chain HXXZ at anisotropy param- eter ∆ = cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar Q-state Potts model at

    √ Q =

    2S + 1 = 2 cosh(λ). The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground states of these two models: dimerization for HAF at all S > 1/2, and Néel order for HXXZ at λ > 0. The results presented include: i) a translation to the above quantum spin systems of the results which were recently proven by Duminil-Copin-Li-Manolescu for a broad class of two-dimensional random-cluster models, and ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray-Spinka of the dis- continuity of the phase transition for Q > 4. Altogether, the quantum manifestation of the change between Q = 4 and Q > 4 is a transition from a gapless ground state to a pair of gapped and extensively distinct ground states.

    1 Introduction

    The focus of this work is the structure of the ground states in two families of antiferromagnetic quantum spin chains, each of which includes the spin-1/2 Heisenberg anti-ferromagnet as a special case. In the infinite volume limit, with the exception of their common root, in both cases the systems exhibit symmetry breaking at the level of ground states. The physics underlying the phenomenon is different. In one case it is extensive quantum frustration which causes dimerization expressed here in spacial energy oscillations. In the other case, the Hamiltonian is frustration free, and the symmetry breaking is expressed in long-range Néel order. Yet, in mathematical terms both phenomena are analyzable through a common random loop representation. Curiously, a similar loop system appears also as the auxiliary scaffoldings of a classical planar Q-state Potts models for which the symmetry breaking relates to a discontinuity in the order parameter.

    The models under consideration have been studied extensively, and hence the specific results we discuss may be regarded as known, at one level or another. The techniques which have been applied for the purpose include numerical works, Bethe ansatz calculations [1, 8, 9, 10, 11, 26], and cluster

    ∗Corresponding author

    1

  • expansions [29]. The validity of Bethe ansatz calculations for similar systems has recently received support through a careful mathematical analysis [19]. The results presented here are based on non- perturbative structural arguments. They may be worth presenting since in the models considered here such arguments allow full characterization of the conditions under which the symmetry breaking occurs, as well as other qualitative features of the model’s grounds states. The relation between the models may be of intrinsic interest. At the mathematical level it plays an essential role in the non-perturbative proof of symmetry breaking which is the main result presented here.

    1.1 Antiferromagnetic SU(2S + 1) invariant spin chains with projection based interaction

    The first family of models concerns the quantum spin chain of spin-S operators Su := (S x u , S

    y u, Szu)

    with the antiferromagnetic Hamiltonian

    H (L) AF =

    L−1∑ u=−L+1

    [2Su · Su+1 − 1/2] for S = 1/2,−(2S + 1)P (0)u,u+1 for S ≥ 1/2, (1.1) with P

    (0) u,u+1 the orthogonal projection onto the singlet state of spins on neighboring sites u, (u+1) ∈

    ΛL := {−L+1, . . . , L}. When referring to such local operators in the context of the product Hilbert space HL :=

    ⊗L v=−L+1 C2S+1, they are to be understood as acting as the identity on the other

    components of the product. This model was studied by Affleck [1], Batchelor and Barber [8, 9], Klümper [26], Aizenman and Nachtergaele [5], and more recently Ueltschi and Nachtergaele [29].

    For S = 1/2, in which case the two formulations agree, this system is just the Heisenberg anti- ferromagnet, for which the L = 1 ground state P (0) = |D1〉〈D1| is the maximally entangled state

    |D1〉 := (|+,−〉 − |−,+〉)/ √

    2 . (1.2)

    The more general expression of the rank-one projection is:

    P (0)u,v := 1 [|SSSu +SSSv| = 0] = 1

    2s+ 1

    S∑ m,m′=−S

    (−1)m−m′ ∣∣m,−m〉u,v〈m′,−m′∣∣ , (1.3)

    where |m〉u are the eigenstates of Szu, for which Szu|m〉u = m|m〉u, m ∈ {−S,−S+ 1, . . . , S})1. Here and in the following, the subscripts on the vectors indicate on which tensor component of HL they act.

    Each of the two-spin interaction terms in (1.1) is minimized in the state in which the two spins are coherently intertwined into the unique state in which |Su+Sv| = 0. However, a spin cannot be locked into such a state with both its neighbours simultaneously. This effect, which results in the spin- Peierls instability, is purely quantum as there is no such restriction for classical spins. Classical spin models exhibit frustration when placed on a non-bipartite graph with antiferromagnetic interactions, and also on arbitrary graphs under suitably mixed interactions. Such geometric frustration is then shared by their quantum counterparts.

    The naive pairing depicted in Fig. 1 suggests that in finite volume the ground-states’ local energy density may not be homogeneous and have a bias triggered by the boundary conditions, i.e. the parity

    1The projection P (0) u,v can also be expressed as a polynomial of degree 2S in SSSu ·SSSv, for instance P (0)u,v = −SSSu ·SSSv+1/4

    for S = 1/2 and P (0) u,v = 1− (SSSu ·SSSv)2 for S = 1.

    2

  • Figure 1: The natural pairing in ΛL = {−L + 1, . . . , L − 1, L} for L = 3 and L = 4. Notice the difference at u = 0.

    of L. Indeed, through approximations, numerical simulations, or the probabilistic representation of [5] (our preferred method), one may see that the local energy density of the corresponding finite-

    volume ground states 〈·〉(gs)L is not homogeneous and satisfies

    (−1)L [ 〈P2n,2n+1〉(gs)L − 〈P2n−1,2n〉

    (gs) L

    ] > 0 . (1.4)

    An interesting question is whether this bias persists in the limit L → ∞, in which case in the infinite-volume limit the system has (at least) two distinct ground states, for which the expectation values of local observables F are given by

    〈F 〉even := lim L→∞ Leven

    〈F 〉(gs)L and 〈F 〉odd := limL→∞ Lodd

    〈F 〉(gs)L , (1.5)

    where the limit is interpreted in the weak sense, i.e. with F being any (fixed) local bounded operator. These are generated by products of spin operators

    FU := k∏ j=1

    S αj uj , uj ∈ U , αj ∈ {x, y, z} (1.6)

    which are supported in some bounded set U ⊂ Z. In finite-volume, their (imaginary) time-evolved counterparts are given by

    F (L) U (t) := e

    −tH(L)AF FU e tH

    (L) AF .

    The corresponding truncated correlations also converge, e.g. for any fixed t ∈ R,

    〈FU (t);FV 〉even := lim L→∞ Leven

    〈F (L)U (t)FV 〉 (gs) L − 〈F

    (L) U (t)〉

    (gs) L 〈FV 〉

    (gs) L , (1.7)

    and similarly for 〈FU (t);FV 〉odd. The separate convergence of the limits (1.5) or (1.7) was established in [5] through probabilistic

    techniques which are enabled by the loop representation which is presented below. This represen- tation also led to the following dichotomy.2

    Proposition 1.1 (cf. Thm. 6.1 in [5]). For each value of S ∈ N/2 one of the following holds true:

    1. The two ground states 〈·〉even and 〈·〉odd are distinct, each invariant under the 2-step shift, each being the 1-step shift of the other. Furthermore, their translation symmetry breaking is manifested in energy oscillations, namely, for every n ≥ N

    〈P (0)2n,2n+1〉even − 〈P (0) 2n−1,2n〉even > 0 . (1.8)

    2This version of the AN dichotomy is a bit more carefully crafted than in the original work, as the two options stated there need not be mutually exclusive. However, as (1.10) shows, ipso-facto they are.

    3

  • 2. The even and odd ground states coincide, and form a translation invariant ground state 〈·〉 with slowly decaying correlations, satisfying∑

    v∈Z |v| |〈SSS0 ·SSSv〉| =∞ . (1.9)

    For S = 1/2 the second alternative is known to hold [20]. In this case the model reduces to the quantum Heisenberg antiferromagnet, for which Bethe [7] predicted the low-energy spectrum exactly by means of his famous ansatz. In the converse direction, dimerization in this model was established for S ≥ 8 [29] through a cluster expansion. The gap between these results is closed here through a structural proof that for all S > 1/2 the first option holds (regardless of the parity of 2S).

    Theorem 1.2. For all S > 1/2:

    1. the even and odd ground-states, defined by (1.5), differ. They are translates of each other, and exhibit the energy oscillation (1.8).

    2. there exist C = C(S), ξ = ξ(S) < ∞ such that for all U, V ⊂ Z with distance dist(U, V ) and any t ∈ R:

    |〈FU (t);FV 〉even| ≤ C e−(dist(U,V )+|t|)/ξ . (1.10)

    The proof draws on the progress which was recently made in the study of the related loop models. In [19], the loop represe