arXiv subjects

Erik Luijten

Publications and source records attributed to Erik Luijten.

52 records · Page 3Linked to original sources

arXiv2000

Comment on ``Scaling Laws for a System with Long-Range Interactions within Tsallis Statistics''

In their recent Letter [Phys. Rev. Lett. 83, 4233 (1999)], Salazar and Toral (ST) study numerically a finite Ising chain with non-integrable interactions decaying like 1/r^(d+sigma) where -d <= sigma <= 0 (like ST, we discuss general dimensionality d). In particular, they explore a presumed connection between non-integrable interactions and Tsallis's non-extensive statistics. We point out that (i) non-integrable interactions provide no more motivation for Tsallis statistics than do integrable interactions, i.e., Gibbs statistics remain meaningful for the non-integrable case, and in fact provide a {\em complete and exact treatment}; and (ii) there are undesirable features of the method ST use to regulate the non-integrable interactions.

cond-mat.stat-mech↗

arXiv1999

Test of renormalization predictions for universal finite-size scaling functions

We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously has been found for short-range interactions, this leads to a singular epsilon-expansion, where epsilon is the distance to the upper critical dimension. Subsequently, we check the results by numerical simulations of spin models in the same universality class. Our systems offer the essential advantage that epsilon can be varied continuously, allowing an accurate examination of the region where epsilon is small. The numerical calculations turn out to be in striking disagreement with the predicted singularity.

cond-mat.stat-mech↗

arXiv1999

Optimized energy calculation in lattice systems with long-range interactions

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blöte algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N^2) problem for systems of size N. We show how this can be reduced to an O(N logN) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.

cond-mat.stat-mech↗

arXiv1999

Monte Carlo investigations of phase transitions: status and perspectives

Using the concept of finite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies of Ising models are discussed, as well as the extension to models of polymer mixtures and solutions. It is shown that using appropriate cluster algorithms, even the scaling functions describing the crossover from the Ising universality class to the mean-field behavior with increasing interaction range can be described. Additionally, the issue of finite-size scaling in Ising models above the marginal dimension (d*=4) is discussed.

cond-mat.stat-mech↗

arXiv1999

Monte Carlo simulation of spin models with long-range interactions

An efficient Monte Carlo algorithm for the simulation of spin models with long-range interactions is discussed. Its central feature is that the number of operations required to flip a spin is independent of the number of interactions between this spin and the other spins in the system. In addition, critical slowing down is strongly suppressed. In order to illustrate the range of applicability of the algorithm, two specific examples are presented. First, some aspects of the Kosterlitz--Thouless transition in the one-dimensional Ising chain with inverse-square interactions are calculated. Secondly, the crossover from Ising-like to classical critical behavior in two-dimensional systems is studied for several different interaction profiles.

cond-mat.stat-mech↗

arXiv1999

Do crossover functions depend on the shape of the interaction profile?

We examine the crossover from classical to non-classical critical behaviour in two-dimensional systems with a one-component order parameter. Since the degree of universality of the corresponding crossover functions is still subject to debate, we try to induce non-universal effects by adding interactions with a second length scale. Although the crossover functions clearly depend on the range of the interactions, they turn out to be remarkably robust against further variation of the interaction profile. In particular, we find that the earlier observed non-monotonic crossover of the effective susceptibility exponent occurs for several qualitatively different shapes of this profile.

cond-mat.stat-mech↗

arXiv1999

Critical properties of the three-dimensional equivalent-neighbor model and crossover scaling in finite systems

Accurate numerical results are presented for the three-dimensional equivalent-neighbor model on a cubic lattice, for twelve different interaction ranges (coordination number between 18 and 250). These results allow the determination of the range dependences of the critical temperature and various critical amplitudes, which are compared to renormalization-group predictions. In addition, the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin-1/2 model and confirms earlier conclusions that the leading Wegner correction must be negative for the three-dimensional (nearest-neighbor) Ising model. By complementing these results with Monte Carlo data for systems with coordination numbers as large as 52514, the full finite-size crossover curves between classical and Ising-like behavior are obtained as a function of a generalized Ginzburg parameter. Also the crossover function for the effective magnetic exponent is determined.

cond-mat.stat-mech↗

arXiv1999

Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model

Monte Carlo results for the moments of the magnetization distribution of the nearest-neighbor Ising ferromagnet in a L^d geometry, where L (4 \leq L \leq 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d=5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called ``lowest-mode'' theory, which uses a single scaling variable tL^{d/2} where t=T/T_c-1 is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the ``lowest-mode'' results (to which the CD theory can easily be reduced in the limit t \to 0, L \to \infty, tL^{d/2} fixed) for the fourth-order cumulant, discrepancies are found for the susceptibility (L^d ). Reasons for these problems are briefly discussed.

cond-mat.stat-mech↗

arXiv1998

Nature of crossover from classical to Ising-like critical behavior

We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility to vary the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semi-phenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition we present strong evidence that the exponent relations do not hold between effective exponents.

cond-mat.stat-mech↗

arXiv1997

Crossover scaling in two dimensions

We determine the scaling functions describing the crossover from Ising-like critical behavior to classical critical behavior in two-dimensional systems with a variable interaction range. Since this crossover spans several decades in the reduced temperature as well as in the finite-size crossover variable, it has up to now largely evaded a satisfactory numerical determination. Using a new Monte Carlo method, we could obtain accurate results for sufficiently large interactions ranges. Our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal. Also the so-called effective exponents are discussed and we show that these can vary nonmonotonically in the crossover region.

cond-mat.stat-mech↗

arXiv1997

Nonmonotonical crossover of the effective susceptibility exponent

We have numerically determined the behavior of the magnetic susceptibility upon approach of the critical point in two-dimensional spin systems with an interaction range that was varied over nearly two orders of magnitude. The full crossover from classical to Ising-like critical behavior, spanning several decades in the reduced temperature, could be observed. Our results convincingly show that the effective susceptibility exponent gamma_eff changes nonmonotonically from its classical to its Ising value when approaching the critical point in the ordered phase. In the disordered phase the behavior is monotonic. Furthermore the hypothesis that the crossover function is universal is supported.

cond-mat.stat-mech↗

arXiv1997

Classical critical behavior of spin models with long-range interactions

We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we give, to our knowledge, the first results for the critical couplings.

cond-mat.stat-mech↗

arXiv1997

Universality and the five-dimensional Ising model

We solve the long-standing discrepancy between Monte Carlo results and the renormalization prediction for the Binder cumulant of the five-dimensional Ising model. Our conclusions are based on accurate Monte Carlo data for systems with linear sizes up to L=22. A detailed analysis of the corrections to scaling allows the extrapolation of these results to L=\infinity. Our determination of the critical point, K_c=0.1139150 (4), is more than an order of magnitude more accurate than previous estimates.

cond-mat.stat-mech↗

arXiv1996

Medium-range interactions and crossover to classical critical behavior

We study the crossover from Ising-like to classical critical behavior as a function of the range R of interactions. The power-law dependence on R of several critical amplitudes is calculated from renormalization theory. The results confirm the predictions of Mon and Binder, which were obtained from phenomenological scaling arguments. In addition, we calculate the range dependence of several corrections to scaling. We have tested the results in Monte Carlo simulations of two-dimensional systems with an extended range of interaction. An efficient Monte Carlo algorithm enabled us to carry out simulations for sufficiently large values of R, so that the theoretical predictions could actually be observed.

cond-mat↗

arXiv1996

Finite-size Scaling and Universality above the Upper Critical Dimensionality

According to renormalization theory, Ising systems above their upper critical dimensionality d_u = 4 have classical critical behavior and the ratio of magnetization moments Q = ^2 / has the universal value 0.456947... However, Monte Carlo simulations of d = 5 Ising models have been reported which yield strikingly different results, suggesting that the renormalization scenario is incorrect. We investigate this issue by simulation of a more general model in which d_u < 4, and a careful analysis of the corrections to scaling. Our results are in a perfect agreement with the renormalization theory and provide an explanation of the discrepancy mentioned.

cond-mat↗

arXiv1995

Ising Universality in Three Dimensions: A Monte Carlo Study

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbor interactions, a spin-1/2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y_t = 1.587 (2), y_h = 2.4815 (15) and y_i = -0.82 (6). The universal ratio Q = ^2/ is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbor spin-1/2 model is K_c=0.2216546 (10).

cond-mat↗