Abstract. -We study the classical 120-degree and related orbital models. These are the classical limits of quantum models which describe the interactions among orbitals of transitionmetal compounds. We demonstrate that at low temperatures these models exhibit a long-range order which arises via an "order by disorder" mechanism. This strongly indicates that there is orbital ordering in the quantum version of these models, notwithstanding recent rigorous results on the absence of spin order in these systems.Introduction. -The properties of transition-metal (TM) compounds are a topic of longstanding interest. In these materials, the fractional filling of the 3d-shells in the TM ion provides a novel facet: The splitting of the t 2g and e g orbitals by the crystal field can produce situations with a single dynamical electron (or a hole) on each site along with multiple orbital degrees of freedom [1][2][3] [7]. The presence of the extra degrees of freedom raises the theoretical possibility of global, cooperative effects; i.e., orbital ordering. Such ordering may be observed via associated orbital-related magnetism and lattice distortions or, e.g., by resonant X-ray scattering techniques in which the 3d orbital order is detected by its effect on excited 4p states [8].The case for orbital ordering has been bolstered by detailed calculations and various other considerations [9]. However, alternate perspectives and various conceptual doubts have been raised concerning the entire picture of long-range orbital ordering [10,11]. In particular, at the theoretical level, a satisfactory justification of orbital ordering has not yet been provided [12]. The goal of this Letter is to present arguments which irrefutably demonstrate that orbital ordering indeed occurs. We will discuss primarily the so-called 120• -model which describes the situations when the e g orbitals are occupied by a single electron.

A new cluster algorithm based on invasion percolation is described. The algorithm samples the critical point of a spin system without a priori knowledge of the critical temperature and provides an efficient way to determine the critical temperature and other observables in the critical region. The method is illustrated for the two-and three-dimensional Ising models. The algorithm equilibrates spin configurations much faster than the closely related Swendsen-Wang algorithm. PACS numbers: 75.40.Mg, 75. 10.Hk Enormous improvements in simulating systems near critical points have been achieved by using cluster algorithms [1,2]. In the present paper we describe a new cluster method which has the additional property of "selforganized criticality. " In particular, the method can be used to sample the critical region of various spin models without the need to fine tune any parameters (or know them in advance). Here, as in other cluster algorithms, bond clusters play a pivotal role in a Markov process, where successive spin configurations are generated using the Fortuin-Kasteleyn [3] representation to identify clusters of spins for flipping. However, the clusters themselves are identified using invasion percolation. The new algorithm is closely related to the Swendsen-Wang (SW) algorithm [1] and may be adapted for a wide range of systems. For purposes of illustration, in this work we will consider the Ising model. Let us first recall the SW algorithm as applied to an Ising system (in the Potts representation).Starting from a spin configuration, satisfied bonds those connecting spins that are of the same type are occupied with probability, p(p) = 1 -e~where p = J/kttT is the coupling strength. Unsatisfied bonds are never occupied.Clusters of sites connected by occupied bonds are locked into the same spin type, and all clusters (including isolated sites) are independently flipped with probability 1/2. The SW algorithm samples the canonical ensemble of the spin system at coupling P and/or the random cluster (bond configuration) ensemble with parameter p. At T, the. SW algorithm is far more efficient than single spin-Rip methods, because the flipped clusters are also critical droplets [4].Here we propose using invasion percolation [5 -10] to generate the bond clusters for the spin Hips. In the usual invasion percolation, random numbers are independently assigned to the bonds of the lattice. Growth starts from one or more seed sites, and at each step the clusters grow by the addition of the perimeter bond with the smallest random number.If a single cluster grows indefinitely on an infinite lattice, its large scale behavior is presumed to be that of the "incipient infinite cluster" of ordinary percolation. In particular, the fraction of perimeter bonds accepted into the growing cluster approaches the percolation threshold p, . [9,10]. Invasion percolation is thus a self-organized critical phenomenon.For the present, we modify invasion percolation in two ways. First, we initiate cluster growth at all lattice site». Consider this ...

The invaded cluster algorithm, a new method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models.Comment: 39 pages, revtex, 15 figures available on request from machta@phast.umass.edu, to appear in Phys. Rev.

Abstract:We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z d . 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 meanfield 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 Z d 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.

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