自对耦无序分布随机链Potts模型的临界普适性研究

以蒙特卡罗模拟方法对自对耦分布二维随机链q态Potts模型的短时临界行为进行了数值研究.利用初始非平衡演化阶段存在的普适幂指数和有限体积标度行为，数值模拟了在不同形式随机分布时q＝3和q＝8态Potts模型磁临界指数η和动力学临界指数z.计算结果发现η不依赖于自对偶无序分布的具体形式， 从而以数值方法给出了一个关于淬火掺杂自旋系统的临界普适行为的验证. 关键词： 随机链Potts模型 动力学蒙特卡罗模拟 临界普适性     

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

The phase transitions and critical phenomena in the three-dimensional (3D) site-diluted q-state Potts models on a simple cubic lattice are explored. We systematically study the phase transitions of the models for q=3 and q=4 on the basis of Wolff high-effective algorithm by the Monte–Carlo (MC) method. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.00–0.65. It is shown that introducing of weak disorder (p∼0.95) into the system is sufficient to change the first order phase transition into a second order one for the 3D 3-state Potts model, while for the 3D 4-state Potts model, such a phase transformation occurs when introducing strong disorder (p∼0.65). Results for 3D pure 3-state and 4-state Potts models (p=1.00) agree with conclusions of mean field theory. The static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the exponent of the correlation radius ν are calculated for the samples on the basis of finite-size scaling theory.     

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.     

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.     

Study of cluster fluctuations in the two-dimensional q-state Potts model

The two-dimensional Potts model with 2 to 10 states is studied using a cluster algorithm to calculate fluctuations in cluster size as well as commonly used quantities like equilibrium averages and the histograms for energy and the order parameter. Results provide information about the variation of cluster sizes depending on the temperature and the number of states. They also give evidence for first-order transition when energy and the order parameter related measurables are inconclusive on small size lattices.     

Dynamic critical behavior of a Swendsen-Wang-Type algorithm for the Ashkin-Teller model

We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (_int. const xC _H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio _int./C _H appears to tend to infinity either as a logarithm or as a small power (0.05p0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.     

Investigating the nonequilibrium aspects of long-range Potts model: Refinement of critical temperatures and raw exponents

In this work, we analyze the q-state Potts model with long-range interactions through nonequilibrium scaling relations commonly used when studying short-range systems. We determine the critical temperature via an optimization method for short-time Monte Carlo simulations. The study takes into consideration two different boundary conditions and three different values of range parameters of the couplings. We also present estimates of some critical exponents, named as raw exponents for systems with long-range interactions, which confirm the non-universal character of the model. Finally, we provide some preliminary results addressing the relations between the raw exponents and the exponents obtained for systems with short-range interactions. The results assert that the methods employed in this work are suitable to study the considered model and can easily be adapted to other systems with long-range interactions.     

Potts Model on Maple Leaf Lattice with Pure Three-Site Interaction

We use Monte Carlo method to study three-state Potts model on maple leaf lattice with pure three-site interaction. The critical behavior of both ferromagnetic and antiferromagnetic cases is studied. Our results confirm that the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. For the antiferromagnetic case the transition is of the first order. We have calculated the energy jump and critical temperature in this area. We find there is a tricritical point separating the first order and second order phases for this system.     

Corner exponents in the two-dimensional Potts model

The critical behavior at a corner in two-dimensional Ising and three-state Potts models is studied numerically on the square lattice using transfer operator techniques. The local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite-size scaling results at the critical point for isotropic or anisotropic couplings. The scaling dimensions compare quite well with the values expected from conformal invariance, provided the opening angle is replaced by an effective one in anisotropic systems.     

Finite-Size Scaling in the p-State Mean-Field Potts Glass: A Monte Carlo Investigation

The p-state mean-field Potts glass with bimodal bond distribution (±J) is studied by Monte Carlo simulations, both for p = 3 and p = 6 states, for system sizes from N = 5 to N = 120 spins, considering particularly the finite-size scaling behavior at the exactly known glass transition temperature T _c. It is shown that for p = 3 the moments q ^(k) of the spin-glass order parameter satisfy a simple scaling behavior, being the appropriate scaling function and T the temperature. Also the specific heat maxima have a similar behavior, , while moments of the magnetization scale as . The approach of the positions T _max of these specific heat maxima to T _c as N is nonmonotonic. For p = 6 the results are compatible with a first-order transition, q ^(k) (q _jump)^k as N but since the order parameter q _jump at T _c is rather small, a behavior q ^(k) N ^-k/3 as N also is compatible with the data. Thus no firm conclusions on the finite-size behavior of the order parameter can be drawn. The specific heat maxima c _V ^max behave qualitatively in the same way as for p = 3, consistent with the prediction that there is no latent heat. A speculative phenomenological discussion of finite-size scaling for such transitions is given. For small N (N 15 for p = 3, N 12 for p = 6) the Monte Carlo data are compared to exact partition function calculations, and excellent agreement is found. We also discuss ratios , for various quantities X, to test the possible lack of self-averaging at T _c.     

Monte Carlo study of tricritical dynamics in two dimensions

The dynamical tricritical behavior for the spin-1 Ising model with single-ion interaction is investigated in two dimensions using Monte Carlo simulations. The nonlinear dynamical tricritical exponentz _t is determined from the asymptotic power-law relaxation of the magnetization. The valuez _t = 1.99 ± 0.04 reported here is the first estimate of the dynamical exponent at a multicritical point, in two dimensions.On leave from Departamento de Fisica, Universidade Federal de Pernambuco, Recife-PE, Brazil.     

Height Representation,Critical Exponents,and Ergodicity in the Four-State Triangular Potts Antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a (2+2)-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagomé lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground-state ensemble of the model. We find that the exponents governing the spin–spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang–Swendsen–Kotecký cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodicity, we perform our simulations on both the torus and the projective plane.     

Dynamic critical behavior of the Swendsen-Wang algorithm: The two-dimensional three-state Potts model revisited

We investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions. Two methods are used: (1) A stochastic optimization process of the Robbins-Monro type, which is designed solely to locate the maximum value of the specific heat; and (2) umbrella sampling, which is designed to sample data over a wide temperature range, as well as to combat the quasiergodicity of Metropolis algorithms in the collapsed phase. We find good evidence that the transition is second order with a divergent specific heat, and that the divergence of the specific heat coincides with the metric collapse.     

Dynamics of the infinite-ranged Potts model

We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.     

Monte Carlo studies of percolation phenomena for a simple cubic lattice

The site-percolation problem on a simple cubic lattice is studied by the Monte Carlo method. By combining results for periodic lattices of different sizes through the use of finite-size scaling theory we obtain good estimates forp _c (0.3115±0.0005), (0.41±0.01), (1.6±0.1), and(0.8±0.1). These results are consistent with other studies. The shape of the clusters is also studied. The average surface area for clusters of sizek is found to be close to its maximal value for the low-concentration region as well as for the critical region. The percentage of particles in clusters of different sizesk is found to have an exponential tail for large values ofk forP <p _c. Forp >p _c there is too much scatter in the data to draw firm conclusions about the size distribution.Work supported in part by USAFOSR Grant #73-2430B and by ERDA #E(11-1)-3077.     

Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

Short-Time Dynamics of Random-Bond Potts Ferromagnet with Continuous Self-Dual Quenched Disorders

We present our Monte Carlo results of the random-bond Potts ferromagnet with the Olson-Young self-dual distribution of quenched disorders in two dimensions. By exploring the short-time scaling dynamics, we find the universal power-law critical behavior of the magnetization and Binder cumulant at the critical point, and thus obtain estimates of the dynamic exponent z and magnetic exponent η, as well as the exponent θ. Our special attention is paid to the dynamic process for the q = 8 Potts model.     

Universal critical properties of the Eulerian bond-cubic model

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations,using an efficient cluster algorithm and a finite-size scaling analysis.The critical points and four critical exponents of the model are determined for several values of n.Two of the exponents are fractal dimensions,which are obtained numerically for the first time.Our results are consistent with the Coulomb gas predictions for the critical O(n) branch for n < 2 and the results obtained by previous transfer matrix calculations.For n=2,we find that the thermal exponent,the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model.These results confirm that the cubic anisotropy is marginal at n=2 but irrelevant for n < 2.     

Logarithmic corrections to finite-size scaling in the four-state Potts model

The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz equations for equivalent eigenstates of a modifiedXXZ chain. Scaled gaps are found to behave for large chain lengthL asx+dL+0[(lnL)^–1], wherex is the anomalous dimension of the associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudesd are in agreement with predictions of conformai invariance. The implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x andd are found to be extremely difficult even with data available from large lattices,L500.