Absence of Magnetism in Continuous-Spin Systems with Long-Range Antialigning Forces

We consider continuous-spin models on the d-dimensional hypercubic lattice with the spins σ _x a priori uniformly distributed over the unit sphere in ℝ ⁿ (with n≥2) and the interaction energy having two parts: a short-range part, represented by a potential Φ, and a long-range antiferromagnetic part λ|x−y|^−s σ _x ⋅σ _y for some exponent s>d and λ≥0. We assume that Φ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For d≥1, s∈(d,d+2] and any λ>0, we then show that the expectation of each σ _x vanishes in all translation-invariant Gibbs states. In particular, the spontaneous magnetization is zero and block-spin averages vanish in all (translation invariant or not) Gibbs states. This contrasts the situation of λ=0 where the ferromagnetic nearest-neighbor systems in d≥3 exhibit strong magnetic order at sufficiently low temperatures. Our theorem extends an earlier result of A. van Enter ruling out magnetized states with uniformly positive two-point correlation functions.     

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤ ^d with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.     

Colligative Properties of Solutions: I. Fixed Concentrations

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent--solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing-point depression. The limit of infinitesimal concentrations is described in a subsequent paper.     

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

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 spread-out 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 mean-field values and the states themselves converge to a product measure.     

Colligative Properties of Solutions: II. Vanishing Concentrations

We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale like c=ξ/L and h=b/L, respectively. We find that there exists a critical value ξt such that no phase separation occurs for ξ≤ξt while, for ξ>ξt, the two phases of the solvent coexist for an interval of values of b. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when b reaches a critical value. For certain values of system parameters, under “frozen” boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing b and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon.     

Orbital Ordering in Transition-Metal Compounds: I. The 120-Degree Model

We study the classical version of the 120-model. This is an attractive nearest-neighbor system in three dimensions with XY (rotor) spins and interaction such that only a particular projection of the spins gets coupled in each coordinate direction. Although the Hamiltonian has only discrete symmetries, it turns out that every constant field is a ground state. Employing a combination of spin-wave and contour arguments we establish the existence of long-range order at low temperatures. This suggests a mechanism for a type of ordering in certain models of transition-metal compounds where the very existence of long-range order has heretofore been a matter of some controversy.© 2005 M. Biskup, L. Chayes and Z. Nussinov. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.Acknowledgement The research of M.B. and L.C. was supported by the NSF under the grant NSF DMS-0306167. Parts of this paper were written when M.B. was visiting Microsoft Research in Redmond whose hospitality is gratefully acknowledged. The authors wish to thank Jeroen van den Brink for discussions and clarifications and two anonymous referees for suggestions that led to improvements in the presentation.