Lincoln Chayes

Avoided critical behavior in a uniformly frustrated system

We study the effects of weak long-ranged antiferromagnetic interactions of strength Q on a spin model with predominant short-ranged ferromagnetic interactions. In three dimensions, this model exhibits an avoided critical point in the sense that the critical temperature Tc(Q = 0) is strictly greater than limQ→0Tc(Q). The behavior of this system at temperatures less than Tc(Q = 0) is controlled by the proximity to the avoided critical point. We also quantize the model in a novel way to study the interplay between charge-density wave and superconducting order.

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

AbstractWe study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L2, inverse temperature β>βc and overall magnetization conditioned to take the value m⋆L2−2m⋆vL, where βc−1 is the critical temperature, m⋆=m⋆(β) is the spontaneous magnetization and vL is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when vL3/2L−2 tends to a definite limit. Specifically, we identify a dimensionless parameter Δ, proportional to this limit, a non-trivial critical value Δc and a function λΔ such that the following holds: For ΔΔc, there is a single, Wulff-shaped droplet containing a fraction λΔ≥λc=2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, λΔ and Δ are related via a universal equation that apparently is independent of the details of the system.

Rigorous Analysis of Discontinuous Phase Transitions via Mean-Field Bounds

Abstract: We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice ℤd. Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean- field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on ℤd undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q≥3, the r-component cubic models with r≥4 and the O(N)-nematic liquid-crystal models with N≥3.

Invaded cluster algorithm for equilibrium critical points.

Cluster algorithms are reviewed and a new approach, the invaded cluster algorithm, is described. Invaded cluster algorithms sample critical points without input of the critical temperature. Instead, the critical temperature is an output of the method. Invaded cluster algorithms have less critical slowing than other cluster methods for the Ising model.

GRAPHICAL REPRESENTATIONS FOR ISING SYSTEMS IN EXTERNAL FIELDS

A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin–Teller) model. Bonds in the graphical representation connect the Ashkin–Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.

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.

Phase transitions in Mandelbrot's percolation process in three dimensions

SummaryWe study the phase structure and transitions in three-dimensional Mandelbrot percolation—a process which generates random fractal sets. We establish the existence of three distinct phase transitions, and we show that two of these transitions, corresponding to percolation across the initial set by paths and sheets, are discontinuous.

Graphical representations and cluster algorithms for critical points with fields

A two-replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic Ising systems with arbitrary fields. Critical points are associated with the percolation threshold of the graphical representation. Results from numerical simulations of the Ising model in a staggered field are presented. The dynamic exponent for the algorithm is measured to be less than 0.5.

Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry

Abstract.We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of “order by disorder,” whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.

PARALLEL INVADED CLUSTER ALGORITHM FOR THE ISING MODEL

A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 40962 and 5123) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482±0.001(β/ν=0.518±0.001) for the three-dimensional Ising model.