882 research outputs found
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-Bl\"ote 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.Comment: 10 pages, including 8 EPS figures. To appear in Phys. Rev. E. Also
available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
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.Comment: Corrected shift of critical temperature and some typos. To appear in
Phys. Rev. E. 18 pages RevTeX, including 10 EPS figures. Also available as
PDF file at http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case
We investigate a two-dimensional Ising model with long-range interactions
that emerge from a generalization of the magnetic dipolar interaction in spin
systems with in-plane spin orientation. This interaction is, in general,
anisotropic whereby in the present work we focus on the isotropic case for
which the model is found to be at its upper critical dimensionality. To
investigate the critical behavior the temperature and field dependence of
several quantities are studied by means of Monte Carlo simulations. On the
basis of the Privman-Fisher hypothesis and results of the renormalization group
the numerical data are analyzed in the framework of a finite-size scaling
analysis and compared to finite-size scaling functions derived from a
Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The
obtained excellent agreement suggests that at least in the present case the
concept of universal finite-size scaling functions can be extended to the upper
critical dimensionality.Comment: revtex4, 10 pages, 5 figures, 1 tabl
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.Comment: 6 pages, including 3 EPS figures. To appear in Phys. Rev. E. Also
available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
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.Comment: 7 pages + 4 PostScript figures. Accepted for publication in
Europhysics Letters. Also available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
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.Comment: 16 pages RevTeX, 8 PostScript figures. Uses epsf.sty. Also available
as PostScript and PDF file at http://www.tn.tudelft.nl/tn/erikpubs.htm
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.Comment: 6 pages LaTeX, 1 PostScript figure. Uses cite.sty (included) and
epsf.sty. Also available as PostScript and PDF file at
http://www.tn.tudelft.nl/tn/erikpubs.htm
Crossover critical behavior in the three-dimensional Ising model
The character of critical behavior in physical systems depends on the range
of interactions. In the limit of infinite range of the interactions, systems
will exhibit mean-field critical behavior, i.e., critical behavior not affected
by fluctuations of the order parameter. If the interaction range is finite, the
critical behavior asymptotically close to the critical point is determined by
fluctuations and the actual critical behavior depends on the particular
universality class. A variety of systems, including fluids and anisotropic
ferromagnets, belongs to the three-dimensional Ising universality class. Recent
numerical studies of Ising models with different interaction ranges have
revealed a spectacular crossover between the asymptotic fluctuation-induced
critical behavior and mean-field-type critical behavior. In this work, we
compare these numerical results with a crossover Landau model based on
renormalization-group matching. For this purpose we consider an application of
the crossover Landau model to the three-dimensional Ising model without fitting
to any adjustable parameters. The crossover behavior of the critical
susceptibility and of the order parameter is analyzed over a broad range (ten
orders) of the scaled distance to the critical temperature. The dependence of
the coupling constant on the interaction range, governing the crossover
critical behavior, is discussedComment: 10 pages in two-column format including 9 figures and 1 table.
Submitted to J. Stat. Phys. in honor of M. E. Fisher's 70th birthda
A Geometrical Interpretation of Hyperscaling Breaking in the Ising Model
In random percolation one finds that the mean field regime above the upper
critical dimension can simply be explained through the coexistence of infinite
percolating clusters at the critical point. Because of the mapping between
percolation and critical behaviour in the Ising model, one might check whether
the breakdown of hyperscaling in the Ising model can also be intepreted as due
to an infinite multiplicity of percolating Fortuin-Kasteleyn clusters at the
critical temperature T_c. Preliminary results suggest that the scenario is much
more involved than expected due to the fact that the percolation variables
behave differently on the two sides of T_c.Comment: Lattice2002(spin
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.Comment: 4 pages RevTeX 3.0/3.1, 5 Encapsulated PostScript figures. Uses
epsf.sty. Accepted for publication in Physical Review Letters. Also available
as PostScript and PDF file at http://www.tn.tudelft.nl/tn/erikpubs.htm
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