Quantum Heisenberg models and their probabilistic representations

Abstract

Abstract. These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops effectively perform a coagulation-fragmentation process. This suggests that their joint distribution is Poisson-Dirichlet. These ideas are far from being proved, but they are backed by several rigorous results, notably of Dyson-Lieb-Simon and Schramm.

Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1quantum spin systems

“…The random interchange model is equivalent to the random loop model of Section IV without the factor 3 |L(ω)| . It was later understood that the Poisson-Dirichlet distribution is also present in three-dimensional systems [13]. This behavior is actually fairly general and concerns many systems where one-dimensional objects have macroscopic size.…”

“…This key property is certainly not obvious and the interested reader is referred to a detailed discussion for lattice permutations with numerical checks [24]. It follows that the lengths of macroscopic loops satisfy an effective split-merge process, and the invariant distribution is Poisson-Dirichlet with parameter θ = r s /r m = 3 [13,26]. The case u ∈ (0, 1) is different because loops split with only half the rate above.…”

“…We restrict ourselves to J 2 0 where the model possesses probabilistic representations; quantum spin correlations are given by certain random loop correlations. In dimensions three and higher, it has been recently understood that the joint distribution of the lengths of long loops takes a universally simple form, the Poisson-Dirichlet distribution [13], and this allows to calculate the expectation of certain long-range two-point functions. They can also be calculated using identities from symmetry breaking.…”

Split-and-Merge in Stationary Random Stirring on Lattice Torus

“…Accordingly, the critical value of time, β c , corresponds, in physical terms, to critical inverse temperature. This is reflected by our choice of notation.Inspired by the exhaustive analysis of the Curie-Weiss mean field version of the problem by Schramm, cf [16], and supported by numerical evidence, a refinement of this conjecture (see [10]) claims that beyond the critical time β c , the macroscopically scaled cycle lengths converge in distribution to the Poisson-Dirichlet law PD(1). For precise formulation see Conjecture 2 in section 1.6 below.The work presented in this note is primarily motivated by the following further refinement of the above conjectures.…”

“…In particular, it turned out that the asymptotics of the cycle structure dynamics of random stirring on the d-dimensional discrete tori T N , as N → ∞, is of paramount importance for understanding the emergence of so-called off-diagonal long range order in the spin-1 2 isotropic quantum Heisenberg ferromagnet (for dimensions d ≥ 3) -a Holy Grail of mathematically rigorous quantum statistical physics. For details, see [19] or the surveys [10,20].The main and best known conjecture in this context (see [19]) states that, for dimensions d ≥ 3, there exists a positive and finite critical time β c = β c (d) beyond which cycles of macroscopic size of the random stirring emerge. For precise formulation see Conjecture 1 in section 1.6 below.Note that in the Feynman-Kac (a.k.a.…”