Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

Abstract

We consider a class of spin systems on Z d with vector valued spins (S x ) that interact via the pair-potentials J x,y S x · S y . The interactions are generally spreadout in the sense that the J x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d ≥ 3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1, 2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique "state," then in any sequence of translation-invariant Gibbs states various observables converge to their meanfield values and the states themselves converge to a product measure.

Reflection Positivity and Phase Transitions in Lattice Spin Models

“…In [9] it has been shown that, for n ≥ 3, the order parameter undergoes a discontinuous transition at intermediate temperatures; van Enter and Shlosman [41,42] later proved such transitions in highly non-linear cases. Similar "mean-field driven" first order phase transitions have also been proved for the cubic model [9] and the Blume-Capel model [10].…”

“…One of the early connections to the models on the complete graph appears in Ellis' textbook on large-deviation theory [40]. Most of this section is based on the papers of Biskup and Chayes [9] and Biskup, Chayes and Crawford [10]. The Key Estimate had been used before in some specific cases; e.g., for the Ising model in the paper by Bricmont, Kesten, Lebowitz and Schonmann [22] and for the q-state Potts model in the paper by Kesten and Schonmann [69].…”

Mean-field theory of the Potts gas

“…over the parameters {x a }, and this minimization problem is well-known from the analysis of the usual mean-field Potts model on the lattice: the case h = 0 was discussed by Wu in [2], and the case h = 0 has recently been studied by Biskup et al [20,21], see also [10]. In the next sections we collect their results and discuss the consequences for the v-dependence of f (v, β) and p(v, β).…”

“…No rigorous results, however, are available on the system's behaviour at the activity threshold to colour-ordering -in fact, even the existence of such a threshold is unknown -, and only numerical results are available [17,18,19]. On the other hand, for a number of lattice models it has recently been shown that an understanding of the mean-field theory may be helpful for proving first-order phase transitions for real systems [20,21]. So, it might be worthwile to clarify also the mean-field theory of the Potts gas, as a first step in understanding the behaviour of the real Potts gas at the activity threshold.…”

A Celebration of Jürg and Tom

“…Remarkably, after more than thirty years, it remains the only method known to rigorously prove breaking of nonabelian symmetry-even for the abelian case, there is only one other approach to the short-range case using multiscale analysis (see below). For slow decay two-dimensional plane rotors, there are also results of .Among later applications of infrared bounds are Sokal's specific heat bounds [59], Aizenman's [1] and Fröhlich's [24] proofs of the triviality of φ 4 theories in five or more dimensions, the Aizenman-Fernández analysis of long-range models [4], Helffer's estimates of eigenvalue splitting for certain Schrödinger operators in the thermodynamic limit [43,44], and the work of Biskup-Chayes on mean-field driven phase transitions [7,8].Reflection positivity was introduced in Euclidean field theory by Osterwalder-Schrader [54] and was a key element, albeit implicitly, Date: September 15, 2008. Mathematics 253-37,…”