69 research outputs found
Crossover phenomena involving the dense O() phase
We explore the properties of the low-temperature phase of the O() loop
model in two dimensions by means of transfer-matrix calculations and
finite-size scaling. We determine the stability of this phase with respect to
several kinds of perturbations, including cubic anisotropy, attraction between
loop segments, double bonds and crossing bonds. In line with Coulomb gas
predictions, cubic anisotropy and crossing bonds are found to be relevant and
introduce crossover to different types of behavior. Whereas perturbations in
the form of loop-loop attractions and double bonds are irrelevant, sufficiently
strong perturbations of these types induce a phase transition of the Ising
type, at least in the cases investigated. This Ising transition leaves the
underlying universal low-temperature O() behavior unaffected.Comment: 12 pages, 8 figure
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.Comment: 24 pages RevTeX 3.0/3.1, including 22 PostScript figures. Uses
epsf.st
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.Comment: 9 pages, 13 Encapsulated PostScript figures. To appear in Eur. Phys.
J. B. Also available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Ising-like transitions in the O() loop model on the square lattice
We explore the phase diagram of the O() loop model on the square lattice
in the plane, where is the weight of a lattice edge covered by a
loop. These results are based on transfer-matrix calculations and finite-size
scaling. We express the correlation length associated with the staggered loop
density in the transfer-matrix eigenvalues. The finite-size data for this
correlation length, combined with the scaling formula, reveal the location of
critical lines in the diagram. For we find Ising-like phase transitions
associated with the onset of a checkerboard-like ordering of the elementary
loops, i.e., the smallest possible loops, with the size of an elementary face,
which cover precisely one half of the faces of the square lattice at the
maximum loop density. In this respect, the ordered state resembles that of the
hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of
represents a softening of its particle-particle potentials. We also
determine critical points in the range . It is found that the
topology of the phase diagram depends on the set of allowed vertices of the
loop model. Depending on the choice of this set, the transition may
continue into the dense phase of the loop model, or continue as a
line of O() multicritical points
Special transitions in an O() loop model with an Ising-like constraint
We investigate the O() nonintersecting loop model on the square lattice
under the constraint that the loops consist of ninety-degree bends only. The
model is governed by the loop weight , a weight for each vertex of the
lattice visited once by a loop, and a weight for each vertex visited twice
by a loop. We explore the phase diagram for some values of . For
, the diagram has the same topology as the generic O() phase diagram
with , with a first-order line when starts to dominate, and an
O()-like transition when starts to dominate. Both lines meet in an
exactly solved higher critical point. For , the O()-like transition
line appears to be absent. Thus, for , the phase diagram displays
a line of phase transitions for . The line ends at in an
infinite-order transition. We determine the conformal anomaly and the critical
exponents along this line. These results agree accurately with a recent
proposal for the universal classification of this type of model, at least in
most of the range . We also determine the exponent describing
crossover to the generic O() universality class, by introducing topological
defects associated with the introduction of `straight' vertices violating the
ninety-degree-bend rule. These results are obtained by means of transfer-matrix
calculations and finite-size scaling.Comment: 19 pages, 11 figure
Single-cluster dynamics for the random-cluster model
We formulate a single-cluster Monte Carlo algorithm for the simulation of the
random-cluster model. This algorithm is a generalization of the Wolff
single-cluster method for the -state Potts model to non-integer values
. Its results for static quantities are in a satisfactory agreement with
those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which
involves a full cluster decomposition of random-cluster configurations. We
explore the critical dynamics of this algorithm for several two-dimensional
Potts and random-cluster models. For integer , the single-cluster algorithm
can be reduced to the Wolff algorithm, for which case we find that the
autocorrelation functions decay almost purely exponentially, with dynamic
exponents , and for , and
4 respectively. For non-integer , the dynamical behavior of the
single-cluster algorithm appears to be very dissimilar to that of the SWCM
algorithm. For large critical systems, the autocorrelation function displays a
range of power-law behavior as a function of time. The dynamic exponents are
relatively large. We provide an explanation for this peculiar dynamic behavior.Comment: 7 figures, 4 table
Equivalent-neighbor percolation models in two dimensions: crossover between mean-field and short-range behavior
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 of equivalent neighbors. We also consider the
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 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 and
Potts models. We determine the renormalization exponent describing a
finite-range perturbation at the mean-field limit as . 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
- …