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 dedicated Monte Carlo method, we could obtain accurate results for sufficiently large interaction 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.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
Finite-size scaling (FSS) for a critical phase transition (
t=0) states that within a window of size
∣t∣∼L−1/ν, the scaling behavior of any observable
Q in a system of linear size
L asymptotically follows a scaling form as
Q(t,L)=LYQQ~(tL1/ν), where
ν is the correlation-length exponent,
YQ is an FSS exponent and
Q~(x) is a function of the scaled distance-to-criticality
x≡tL1/ν. We systematically study the asymptotic scaling behavior of
Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with
t→0 and
∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as
∣t∣∼L−λ with
λ<1/ν, the FSS becomes
λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality
t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
Both first- and second-order transitions get smeared and shifted due to the finite size of a sample. Recent theoretical developments on this finite size rounding of singularities associated with phase transitions are briefly reviewed, and evidence from Monte Carlo simulations for the validity of these concepts will be discussed. It is also shown how finite size scaling theories in conjunction with simulation data from finite systems are a powerful tool for the quantitative study of bulk critical phenomena. We shall also mention extensions such as finite size tests of hyperscaling, finite size scaling of interfacial properties, finite size effects on critical relaxation, etc. A brief discussion of the experimental situation concludes this review.