First-order transitions and thermodynamic properties in the 2D Blume-Capel model: the transfer-matrix method revisited

We investigate the first-order transition in the spin-1 two-dimensional Blume-Capel model in square lattices by revisiting the transfer-matrix method. With large strip widths increased up to the size of 18 sites, we construct the detailed phase coexistence curve which shows excellent quantitative agreement with the recent advanced Monte Carlo results. In the deep first-order area, we observe the exponential system-size scaling of the spectral gap of the transfer matrix from which linearly increasing interfacial tension is deduced with decreasing temperature. We find that the first-order signature at low temperatures is strongly pronounced with much suppressed finite-size influence in the examined thermodynamic properties of entropy, non-zero spin population, and specific heat. It turns out that the jump at the transition becomes increasingly sharp as it goes deep into the first-order area, which is in contrast to the Wang–Landau results where finite-size smoothing gets more severe at lower temperatures.     

A finite-size scaling study of the diamond 3d 3q potts model

We study the 3d 3-state Potts model defined on a diamond lattice (with a coordination number 4). By using finite-size scaling techniques we establish the first-order nature of the phase transition. We study local observables, tunneling events and the correlation length of the system, and we clarify the details of the pseudo-critical behavior which the system undergoes close to the transition region.     

Finite-size scaling of the three-state Potts model on a simple cubic lattice

Finite-size scaling is studied for the three-state Potts model on a simple cubic lattice. We show that the specific heat and the magnetic susceptibility scale accurately as the volume. The correlation length exhibits behaviors expected for a genuine first-order transition; the one extracted from the unsubtracted correlation function shows a characteristic finite-size behavior, whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point.     

First-order transition in a three-dimensional disordered system

We present the first detailed numerical study in three dimensions of a first-order phase transition that remains first order in the presence of quenched disorder (specifically, the ferromagnetic-paramagnetic transition of the site-diluted four states Potts model). A tricritical point, which lies surprisingly near the pure-system limit and is studied by means of finite-size scaling, separates the first-order and second-order parts of the critical line. This investigation has been made possible by a new definition of the disorder average that avoids the diverging-variance probability distributions that plague the standard approach. Entropy, rather than free energy, is the basic object in this approach that exploits a recently introduced microcanonical Monte Carlo method.     

Finite-size effects and scaling for the thermal QCD deconfinementphase transition within the exact color-singlet partition function

We study the finite-size effects for the thermal quantum chromodynamics (QCD) deconfinement phase transition, and use a numerical finite-size scaling analysis to extract the scaling exponents characterizing its scaling behavior when approaching the thermodynamic limit . For this, we use a simple model of coexistence of hadronic gas and color-singlet quark gluon plasma (QGP) phases in a finite volume. The color-singlet partition function of the QGP cannot be exactly calculated and is usually derived within the saddle-point approximation. When we try to do calculations with such an approximate color-singlet partition function, a problem arises in the limit of small temperatures and/or volumes , requiring additional approximations if we want to carry out calculations. We propose in this work a method for an accurate calculation of any quantity of the finite system, without any approximation. By probing the behavior of some useful thermodynamic response functions on the whole range of temperature, it turns out that, in a finite-size system, all singularities in the thermodynamic limit are smeared out and the transition point is shifted away. A numerical finite-size scaling (FSS) analysis of the obtained data allows us to determine the scaling exponents of the QCD deconfinement phase transition. Our results expressing the equality between their values and the space dimensionality is a consequence of the singularity characterizing a first-order phase transition and agree very well with the predictions of other FSS theoretical approaches to a first-order phase transition and with the results of calculations using Monte Carlo methods in both lattice QCD and statistical physics models. Received: 11 January 2005, Revised: 7 July 2005, Published online: 30 August 2005     

Generic Two-Phase Coexistence and Nonequilibrium Criticality in a Lattice Version of Schlögl’s Second Model for Autocatalysis

A two-dimensional atomistic realization of Schlögl’s second model for autocatalysis is implemented and studied on a square lattice as a prototypical nonequilibrium model with first-order transition. The model has no explicit symmetry and its phase transition can be viewed as the nonequilibrium counterpart of liquid-vapor phase separations. We show some familiar concepts from study of equilibrium systems need to be modified. Most importantly, phase coexistence can be a generic feature of the model, occurring over a finite region of the parameter space. The first-order transition becomes continuous as a temperature-like variable increases. The associated critical behavior is studied through Monte Carlo simulations and shown to be in the two-dimensional Ising universality class. However, some common expectations regarding finite-size corrections and fractal properties of geometric clusters for equilibrium systems seems to be inapplicable.     

Three-dimensional 3-state Potts model revisited with new techniques

We report a fairly detailed finite-size scaling analysis of the first-order phase transition in the three-dimensional 3-state Potts model on cubic lattices with emphasis on recently introduced quantities whose infinite-volume extrapolations are governed only by exponentially small terms. In these quantities no asymptotic power series in the inverse volume are involved which complicate the finite-size scaling behaviour of standard observables related to the specific-heat maxima or Binder-parameter minima. Introduced initially for strong first-order phase transitions in q-state Potts models with “large enough” q, the new techniques prove to be surprisingly accurate for a q value as small as 3. On the basis of the high-precision Monte Carlo data of Alves et al. [Phys. Rev. B 43 (1991) 5846], this leads to a refined estimate of β_t = 0.550 565(10) for the infinite-volume transition point.     

Phase transitions of two spin-1/2 Baxter–Wu layers coupled with Ising-type interactions

Using a Monte Carlo simulation and the single histogram reweighting technique,we study the critical behaviors and phase transitions of the Baxter-Wu(BW)model on a two-layer triangular lattice with Ising-type interlayer couplings.Via the finite-size analysis,we obtain the transition temperatures and critical exponents at repulsive and attractive interlayer couplings.The data for the repulsive interlayer coupling suggest continuous transitions,and the critical behaviors are the same as those of the 2D BW model,belonging to the four-state Potts universality class.The reduced energy cumulants and the histograms reveal that attractive coupling leads to weak firstorder phase transitions.The pseudocritical exponents with the existence of the interlayer couplings indicate that the first-order transition is very close to the critical point of the 2D standard BW model.     

The nature of explosive percolation phase transition

In this Letter, we show that the explosive percolation is a novel continuous phase transition. The order-parameter-distribution histogram at the percolation threshold is studied in Erd?s-Rényi networks, scale-free networks, and square lattice. In finite system, two well-defined Gaussian-like peaks coexist, and the valley between the two peaks is suppressed with the system size increasing. This finite-size effect always appears in typical first-order phase transition. However, both of the two peaks shift to zero point in a power law manner, which indicates the explosive percolation is continuous in the thermodynamic limit. The nature of explosive percolation in all the three structures belongs to this novel continuous phase transition. Various scaling exponents concerning the order-parameter-distribution are obtained.     

Monte Carlo studies of the square Ising model with next-nearest-neighbor interactions

We apply a new entropic scheme to study the critical behavior of the square-lattice Ising model with nearest- and next-nearest-neighbor antiferromagnetic interactions. Estimates of the present scheme are compared with those of the Metropolis algorithm. We consider interactions in the range where superantiferromagnetic (SAF) order appears at low temperatures. A recent prediction of a first-order transition along a certain range (0.5–1.2) of the interaction ratio (R=J_nnn/J_nn) is examined by generating accurate data for large lattices at a particular value of the ratio (R=1). Our study does not support a first-order transition and a convincing finite-size scaling analysis of the model is presented, yielding accurate estimates for all critical exponents for R=1. The magnetic exponents are found to obey “weak universality” in accordance with a previous conjecture.     

Monte Carlo studies of finite-size effects at first-order transitions

First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.     

Intrinsic finite-size effects in the two-dimensional XY model with irrational frustration

This study investigates in detail the finite-size scaling of the two-dimensional irrationally frustrated XY model. By means of Monte Carlo simulations with entropic sampling, we examine the size dependence of the specific heat, and find remarkable deviation from the conventional finite-size scaling theory, which reveals novel intrinsic finite-size effects. Relaxation dynamics of the system is also considered, and, correspondingly, finite-size scaling of the relaxation time is examined, again giving evidence for the intrinsic finite-size effects and suggesting a zero-temperature glass transition.     

有限尺度下弱相互作用费米气体的热力学性质

基于巨正则系综理论和数值模拟方法，研究有限尺度下弱相互作用费米气体的热力学性质，给出系统低温下的化学势、能量及热容量的解析式，分析弱相互作用、有限尺度效应对系统热力学性质的影响.研究表明，有限尺度和排斥相互作用增大了系统的化学势、能量,吸引相互作用减小了系统的化学势、能量.相互作用受到尺度的调制，尺度变大，相互作用影响变小，相互作用和尺度效应都受到温度的调制，温度升高，相互作用和尺度的影响减小.尺度和相互作用的一级修正对热容量无影响.     

First-order transition features of the triangular Ising model with nearest- and next-nearest-neighbor antiferromagnetic interactions

We implement a new and accurate numerical entropic scheme to investigate the first-order transition features of the triangular Ising model with nearest-neighbor (J_nn) and next-nearest-neighbor (J_nnn) antiferromagnetic interactions in ratio R=J_nn/J_nnn=1. Important aspects of the existing theories of first-order transitions are briefly reviewed, tested on this model, and compared with previous work on the Potts model. Using lattices with linear sizes L=30,40,…,100,120,140,160,200,240,360 and 480 we estimate the thermal characteristics of the present weak first-order transition. Our results improve the original estimates of Rastelli et al. and verify all the generally accepted predictions of the finite-size scaling theory of first-order transitions, including transition point shifts, thermal, and magnetic anomalies. However, two of our findings are not compatible with current phenomenological expectations. The behavior of transition points, derived from the number-of-phases parameter, is not in accordance with the theoretically conjectured exponentially small shift behavior and the well-known double Gaussian approximation does not correctly describe higher correction terms of the energy cumulants. It is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections.     

有限尺度下弱相互作用费米气体的热力学性质

基于巨正则系综理论和数值模拟方法,研究有限尺度下弱相互作用费米气体的热力学性质,给出系统低温下的化学势、能量及热容量的解析式,分析弱相互作用、有限尺度效应对系统热力学性质的影响.研究表明,有限尺度和排斥相互作用增大了系统的化学势、能量,吸引相互作用减小了系统的化学势、能量.相互作用受到尺度的调制,尺度变大,相互作用影响变小,相互作用和尺度效应都受到温度的调制,温度升高,相互作用和尺度的影响减小.尺度和相互作用的一级修正对热容量无影响.     

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.     

Excited-state quantum phase transitions in systems with two degrees of freedom: II. Finite-size effects

This article extends our previous analysis Stránský et al. (2014) of Excited-State Quantum Phase Transitions (ESQPTs) in systems of dimension two. We focus on the oscillatory component of the quantum state density in connection with ESQPT structures accompanying a first-order ground-state transition. It is shown that a separable (integrable) system can develop rather strong finite-size precursors of ESQPT expressed as singularities in the oscillatory component of the state density. The singularities originate in effectively 1-dimensional dynamics and in some cases appear in multiple replicas with increasing excitation energy. Using a specific model example, we demonstrate that these precursors are rather resistant to proliferation of chaotic dynamics.     

Finite-size effects and phase transition in the three-dimensional three-state Potts model

The three-state Potts model in three dimensions is studied by Monte Carlo and finite-size scaling techniques. Using a histogram method recently proposed by Ferrenberg and Swendsen, the finite-size dependence for the maximum of the specific heat is found to scale with the volume of the system, indicating that the phase transition is of first order. The value of the latent heat per spin and the correlation length at the transition are estimated.     

Recognition ability of the fully connected Hopfield neural network under a persistent stimulus field

We investigate the pattern recognition ability of the fully connected Hopfield model of a neural network under the influence of a persistent stimulus field. The model considers a biased training with a stronger contribution to the synaptic connections coming from a particular stimulated pattern. Within a mean-field approach, we computed the recognition order parameter and the full phase diagram as a function of the stimulus field strength h, the network charge α and a thermal-like noise T. The stimulus field improves the network capacity in recognizing the stimulated pattern while weakening the first-order character of the transition to the non-recognition phase. We further present simulation results for the zero temperature case. A finite-size scaling analysis provides estimates of the transition point which are very close to the mean-field prediction.