Forbidden gap argument for phase transitions proved by means of chessboard estimates

Chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest. In this note we show that the method not only produces a point in the phase diagram where more than one Gibbs states coexist, but that it can also be used to rule out the existence of shift-ergodic states that differ significantly from those proved to exist. For models depending on a parameter (say, the temperature), this shows that the values of the conjugate thermodynamic quantity (the energy) inside the "transitional gap" are forbidden in all shift-ergodic Gibbs states. We point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria.Comment: 26 page

A proof of the Gibbs-Thomson formula in the droplet formation regime

We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs-Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general--albeit heuristic--reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.Comment: LaTeX+times; version to appear in J. Statist. Phy

Partition function zeros at first-order phase transitions: Pirogov-Sinai theory

This paper is a continuation of our previous analysis [BBCKK] of partition functions zeros in models with first-order phase transitions and periodic boundary conditions. Here it is shown that the assumptions under which the results of [BBCKK] were established are satisfied by a large class of lattice models. These models are characterized by two basic properties: The existence of only a finite number of ground states and the availability of an appropriate contour representation. This setting includes, for instance, the Ising, Potts and Blume-Capel models at low temperatures. The combined results of [BBCKK] and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class.Comment: 46 pages, 2 figs; continuation of math-ph/0304007 and math-ph/0004003, to appear in J. Statist. Phys. (special issue dedicated to Elliott Lieb

Emergence of long cycles for random interchange process on hypercubes

Motivated by phase transitions in quantum spin models, we study random permutations of vertices (induced by products of uniform independent random transpositions on edges) in the case of high-dimensional hypercubes. We establish the existence of a transition accompanied by emergence of cycles of diverging lengths. (Joint work with Piotr Miłoś and Daniel Ueltschi.)Non UBCUnreviewedAuthor affiliation: University of Warwick and Charles UniversityFacult

A roughening transition indicated by the behaviour of ground states

International audienceOur aim in this contribution is to present some illustrations for the claim that already by looking at the ground states of classical lattice models, one may meet some interesting and non-trivial structures

True nature of long-range order in a plaquette orbital model

We analyze the classical version of a plaquette orbital model that was recently introduced and studied numerically by Wenzel and Janke. In this model, edges of the square lattice are partitioned into x and z types that alternate along both coordinate directions and thus arrange into a checkerboard pattern of x and z plaquettes; classical O(2) spins are then coupled ferromagnetically via their first components over the x edges and via their second components over the z edges. We prove from first principles that, at sufficiently low temperatures, the model exhibits orientational long-range order (OLRO) in one of the two principal lattice directions. Magnetic order is precluded by the underlying symmetries. A similar set of results is inferred also for quantum systems with large spin although the spin-1/2 instance currently seems beyond the reach of rigorous methods. We point out that the Neel order in the plaquette energy distribution observed in numerical simulations is an artifact of the OLRO and a judicious choice of the plaquette energies. In particular, this order seems to disappear when the plaquette energies are adjusted to vanish at the ground-state level. We also discuss the specific role of the underlying symmetries in Wenzel and Janke's simulations and propose an enhanced method of numerical sampling that could in principle significantly increase the speed of convergence