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arXiv:1112.0395 · Europhysics Letters (EPL)

Mean-field models with short-range correlations

M. Ostilli

Published Mar 1, 2012• 4 citations
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Abstract

Given an arbitrary finite dimensional Hamiltonian H 0 , we consider the model H = H 0 +Δ H , where Δ H is a generic fully connected interaction. By using the strong law of large numbers, we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models, the term H 0 gives rise to short-range correlations and, furthermore, when H 0 has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non-uniform external field and finite-size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.

DOI
10.1209/0295-5075/97/50008

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Given an arbitrary finite dimensional Hamiltonian H 0 , we consider the model H = H 0 +Δ H , where Δ H is a generic fully connected interaction. By using the strong law of large numbers, we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models, the term H 0 gives rise to short-range correlations and, furthermore, when H 0 has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non-uniform external field and finite-size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.
Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac limit. If both attractive and repulsive long-range forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the "free" part of the model.
We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie–Weiss with Glauber dynamics) by solving the associated Fokker–Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.
Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.
We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number zzz of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite zzz are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for q=3q=3q=3 and q=4q=4q=4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.
Abstract We considered a long-range Ising model under Glauber dynamics and calculated the difference from the mean-field approximation in a finite-size system using perturbation theory. To deal with the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, we assumed that certain types of extensive properties have a Gaussian distribution, which turned out to be equivalent to the Kirkwood superposition approximation within the range of first-order perturbation. After several calculations, ordinary differential equations that describe the time development of a two-body correlation were derived. This discussion is the generalization of our previous study which developed a similar consideration on the infinite-range Ising model. The results of the calculation fit those of the numerical simulations for the case in which the decay of the interaction was sufficiently slow; however, they exhibited different behaviors when the decay became rapid.
It has previously been pointed out that the coexistence of infinite-range and short-range interactions causes a system to have a phase transition of the mean-field universality class, in which the cluster size is finite even at the critical point. In the present paper, we study this property in a model of bistable molecules, whose size changes depending on the bistable states. The molecules can move in space, interacting via an elastic interaction. It is known that due to the different sizes, an effective long-range interaction between the spins appears, and thus this model has a mean-field type of phase transition. It is found that the scaling properties of the shift of the critical temperature from the pure short-range limit in the model with infinite-range and short-range interactions hold also in the present model, regarding the ratio of the size of the two states as a control parameter for the strength of the long-range interaction. By studying the structure factor, it is shown that the dependence of the cluster size at the critical temperature also shows the same scaling properties as a previously studied model with both infinite-range and short-range interactions. We therefore conclude that these scaling relations hold universally in hybrid models with both short-range and weak long-range interactions.