Estimates Of Critical Exponents Research Articles

Polydispersed rods on the square lattice.

We study the grand-canonical solution of a system of hard polydispersed rods placed on the square lattice using transfer matrix and finite-size scaling calculations. We determine the critical line separating an isotropic from a nematic phase. No second transition to a disordered phase is found at high density, contrary to what is observed in the monodispersed case. The estimates of critical exponents and the central charge on the critical line are consistent with the Ising universality class.

Erratum: “Ground State Phase Diagram of Frustrated S = 1 XXZ Chains: Chiral Ordered Phases” [J. Phys. Soc. Jpn. 69, 259 (2000)

In the paper, it was incorrectly mentioned that the exactdiagonalization calculation for small systems found that the ground state of the model (1.1) belonged to the subspace of S total 1⁄4 0 with even parity.1) (We call this subspace PE below.) While it is the case for the periodic chains2) and for a wide parameter region of the open chains, we have noticed that the ground state with odd parity appears in some narrow parameter regions of the open chains, as shown in Fig. 1 of this erratum. We note that this is a boundary effect in small systems with frustration, and the infinite-system density-matrix renormalization-group (DMRG) calculation for the subspace PE performed in the original paper should capture the groundstate properties in the thermodynamic limit correctly. Indeed, we have newly performed the infinite-system DMRG for the whole subspace of S total 1⁄4 0 including both even and odd parity states (we call this subspace PEO), and obtained essentially the same results as the previous ones for the subspace PE. For example, as for the estimates of jc1 and jc2 shown in Table II, those for 1⁄4 0, 0.2, and 0.4 remain the same, while comparing the results for PE and PEO, we modify the errors of the estimates for 1⁄4 0:6 (0.8) from ðjc1; jc2Þ 1⁄4 ð0:642 0:002; 0:660 0:003Þ [ð0:715 0:003; 0:720 0:002Þ] to ð0:642þ0:004 0:002; 0:660þ0:009 0:003Þ [ð0:715þ0:005 0:003; 0:720þ0:009 0:002Þ]. The other results in the paper, including the phase diagram and the estimates of critical exponents, remain the same within a numerical accuracy. Therefore, our conclusions remain valid. We also correct misprints and typographical errors, which do not affect our conclusions. (i) In Figs. 2(b) and 5, the distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. The label for the x-axis should read r + 1 instead of r. (ii) Equation (3.2) should read

Large-order aspects of the δ expansion in low-dimensional Ising models.

We investigate the large order aspects of the δ expansion under the estimation procession of the critical quantities. As illustrative examples, we revisit the one-dimensional Ising model for the analytic study and the two-dimensional square Ising model in the high-temperature phase for the numerical experiment to large orders. In both models, the proposed fundamental base on which the estimation protocol should be constructed is investigated in detail and confirmed to be valid. In the square lattice model, we present a new protocol for the estimation of critical exponents and temperature.

Relations between short-range and long-range Ising models.

We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions (J∝r{-d-σ}) on both a one-dimensional chain (d=1) and a square lattice (d=2). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters (d,σ) to that of a short-range (SR) model in an equivalent dimension D. We then study the critical behavior of the d=2 LR model close to the lower critical σ, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.

Scaling behavior of the Heisenberg model in three dimensions

We report on extensive numerical simulations of the three-dimensional Heisenberg model and its analysis through finite-size scaling of Lee-Yang zeros. Besides the critical regime, we also investigate scaling in the ferromagnetic phase. We show that, in this case of broken symmetry, the corrections to scaling contain information on the Goldstone modes. We present a comprehensive Lee-Yang analysis, including the density of zeros, and confirm recent numerical estimates for critical exponents.

A generalized method of estimation of critical exponents through an indigenous susceptometry setup for YBCO materials

The real and imaginary parts of complex susceptibility values of bulk ceramic YBCO materials have been extracted at different frequencies and temperatures by measuring impedances using a single-coil swept-ac method. From these susceptibility components, the ac conductivity and its loss angle have been derived numerically. The dynamic and static critical exponents, z and ν obtained from the scaling of conductivity loss angle based on FFH’s vortex glass (VG) model were found to be in agreement with the values obtained by other workers, in different methods characterizing a continuous phase transition and revealing a stronger correlation in HTS materials. To verify their universality, it has been shown that the exponent values support the scaling features of ac-conductivity (or resistivity) very well and can scale susceptibility also at the proximity of critical temperature. It is highly expected that this simpler and straight forward method of extraction and evaluation of these critical exponents discussed here may be applicable to any other High Temperature Superconductors or other complex materials in bulk form.

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m-axial Lifshitz points. We derive the leading non-trivial 1/n correction for the perpendicular correlation-length exponent νL2 and hence several related thermal exponents to order O(1/n). The results are consistent with known large-n expansions for d-dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d⁎=4+m/2 for generic m∈[0,d]. Analytical results are given for the special case d=4, m=1. For uniaxial Lifshitz points in three dimensions, 1/n coefficients are calculated numerically. The estimates of critical exponents at d=3, m=1 and n=3 are discussed.

Magnetic properties and critical behavior of disordered Fe1−xRuxalloys: A Monte Carlo approach

We study the critical behavior of a quenched random-exchange Ising model with competing interactions on a bcc lattice. This model was introduced in the study of the magnetic behavior of Fe(1-x)Ru(x) alloys for ruthenium concentrations x=0%, x=4%, x=6%, and x=8%. Our study is carried out within a Monte Carlo approach, with the aid of a re-weighting multiple histogram technique. By means of a finite-size scaling analysis of several thermodynamic quantities, taking into account up to the leading irrelevant scaling field term, we find estimates of the critical exponents α, β, γ, and ν, and of the critical temperatures of the model. Our results for x=0% are in excellent agreement with those for the three-dimensional pure Ising model in the literature. We also show that our critical exponent estimates for the disordered cases are consistent with those reported for the transition line between paramagnetic and ferromagnetic phases of both randomly dilute and ±J Ising models. We compare the behavior of the magnetization as a function of temperature with that obtained by Paduani and Branco (2008), qualitatively confirming the mean-field result. However, the comparison of the critical temperatures obtained in this work with experimental measurements suggest that the model (initially obtained in a mean-field approach) needs to be modified.

Critical behavior near the paramagnetic to ferromagnetic phase transition temperature in La 0.7Pb 0.05Na 0.25MnO 3

Critical behavior in La 0.7Pb 0.05Na 0.25MnO 3 has been investigated by dc magnetization measurements. Magnetic data indicate that the compound exhibits a continuous (second-order) paramagnetic (PM) to ferromagnetic (FM) phase transition. Estimates of critical exponents yield δ = 4.80 ± 0.01 , γ = 1.296 ± 0.002 and β = 0.344 ± 0.007 (consistent with both the predictions for the three-dimensional-Heisenberg model and with those reported for materials when the FM transition is ascribed to the double exchange (DE) mechanism as a major origin) with T C = 334.54 ± 0.08 . The critical exponent γ is slightly inferior than predicted from the 3D Heisenberg model. Such a difference may be due, within the context of the quenched disorder, to the presence of some alterations of short-range magnetic order of FM clusters in the PM phase. The temperature variation in the effective exponent ( γ e f f ) is similar to those for disordered ferromagnets.

Finite size scaling study of lattice models in the three-dimensional Ising universality class

We simulate the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter D on the simple cubic lattice. We perform a finite size scaling study of lattices of a linear size up to L=360 to obtain accurate estimates for critical exponents. We focus on values of D, where the amplitudes of leading corrections are small. Furthermore we employ improved observables that have a small amplitude of the leading correction. We obtain nu=0.63002(10), eta=0.03627(10) and omega=0.832(6). We compare our results with those obtained from previous Monte Carlo simulations and high temperature series expansions of lattice models, by using field theoretic methods and experiments.

Monte Carlo study of the critical behavior and magnetic properties of La 2/3Ca 1/3MnO 3 thin films

Critical exponents offer important information concerning the interaction mechanisms near the paramagnetic to ferromagnetic transition. In this work a Monte Carlo–Metropolis simulation of the critical behavior in La 2/3Ca 1/3MnO 3 thin films is addressed. Canonical ensemble averages for magnetization per site, magnetic susceptibility and specific heat of stoichiometric manganite within a three-dimensional classical Heisenberg model with nearest magnetic neighbor interactions are computed. The La 2/3Ca 1/3MnO 3 thin films were simulated addressing the thickness influence and thermal dependence. In the model, Mn magnetic ions are distributed on a simple cubic lattice according to the perovskite structure of this manganite. Ferromagnetic coupling for the bonds Mn 3+–Mn 3+( e g – e g ′), Mn 3+–Mn 4+( e g – d 3) and Mn 3+–Mn 4+( e g ′– d 3) were taken into account. On the basis of finite-size scaling theory, our best estimates of critical exponents, linked to the ferromagnetic to paramagnetic transition, for the correlation length, specific heat, magnetization and susceptibility are, respectively: v=0.56±0.01, α=0.16±0.03, β=0.34±0.04 γ and γ=1.17±0.05. These theoretical results are consistent with the Rushbrooke equalitiy α+2 β+ γ=2.

Fractals meet fractals: self-avoiding random walks on percolation clusters

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results yield estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other.

Universality class of the pair contact process with diffusion

The pair contact process with diffusion (PCPD) is studied with a standard Monte Carlo approach and with simulations at fixed densities. A standard analysis of the simulation results, based on the particle densities or on the pair densities, yields inconsistent estimates for the critical exponents. However, if a well-chosen linear combination of the particle and pair densities is used, leading corrections can be suppressed, and consistent estimates for the independent critical exponents delta=0.16(2) , beta=0.28(2) , and z=1.58 are obtained. Since these estimates are also consistent with their values in directed percolation (DP), we conclude that the PCPD falls in the same universality class as DP.

Statistical physics of the melting of inhomogeneous DNA

We studied how the inhomogeneity of a sequence affects the phase transition that takes place at DNA melting. Unlike previous works, which considered thermodynamic quantities averaged over many different inhomogeneous sequences, we focused on precise sequences and investigated the succession of local openings that lead to their dissociation. For this purpose, we performed transfer-integral-type calculations with two different dynamical models: namely, the heterogeneous Dauxois-Peyrard-Bishop model and the model based on finite stacking enthalpies we recently proposed. It appears that, for both models, the essential effect of heterogeneity is to let different portions of the investigated sequences open at slightly different temperatures. Besides this macroscopic effect, the local aperture of each portion indeed turns out to be very similar to that of a homogeneous sequence with the same length. Rounding of each local opening transition is therefore merely a size effect. For the Dauxois-Peyrard-Bishop model, sequences with a few thousand base pairs are still far from the thermodynamic limit, so that it is inappropriate, for this model, to discuss the order of the transition associated with each local opening. In contrast, sequences with several hundred to a few thousand base pairs are pretty close to the thermodynamic limit for the model we proposed. The temperature interval where a power law holds is consequently broad enough to enable the estimation of critical exponents. On the basis of the few examples we investigated, it seems that, for our model, disorder does not necessarily induce a decrease of the order of the transition.

Estimation of critical exponents from cluster coefficients and the application of this estimation to hard spheres

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix R(mn), whose elements converge to two constants with 1/n(2) correction. We find exact expressions in terms of these correction terms for the two critical exponents describing the density near the two singular termination points of the fluid phase. We apply the method to the hard-spheres model and find that the metastable fluid phase terminates at rho(t) = 0.751[5]. The density near the transition is given by rho(t)-rho approximately (z(t) - z)(sigma'), where the critical exponent is predicted to be sigma' = 0.0877[25]. Interestingly, the termination density is close to the observed glass transition; thus, the above critical behavior is expected to be associated with the onset of glassy behavior in hard spheres.

Ambiguities in estimates of critical exponents for long-range dependent processes

In this article, we show that the Hurst parameter and a well-known wavelet estimator fail to distinguish on/off models with substantially different distributions of on-times. We identify the problem as arising when standard power spectrum techniques, which are well-defined for commonly considered processes, are applied to processes with extremely heavy tailed on-times. We provide an elementary mechanism whereby such extremely heavy-tailed on-time processes arise in a simple queuing model.

Critical exponents for the restricted sandpile

I report large-scale Monte Carlo studies of a one-dimensional height-restricted stochastic sandpile using the quasistationary simulation method. Results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems, but are consistent with recent predictions derived from a Langevin equation for stochastic sandpiles [Ramasco, Phys. Rev. E 69, 045105(R) (2004)]. This suggests that apparent violations of universality in one-dimensional sandpiles are due to strong corrections to scaling and finite-size effects.

Towards a fully automated computation of RG functions for the three-dimensionalO(N) vector model: parametrizing amplitudes

Within the framework of field-theoretical description of second-order phase transitions via the three-dimensionalO(N) vector model, accurate predictions for critical exponents can be obtained from(resummation of) the perturbative series of renormalization-group functions, which are inturn derived—following Parisi’s approach—from the expansions of appropriate fieldcorrelators evaluated at zero external momenta.Such a technique was fully exploited30 years ago in two seminal works of Baker, Nickel, Green and Meiron, which led to the knowledge ofthe β-function up to the six-loop level; they succeeded in obtaining a precise numericalevaluation of all needed Feynman amplitudes in momentum space by lowering thedimensionalities of each integration with a cleverly arranged set of computationalsimplifications. In fact, extending this computation is not straightforward, due both to thefactorial proliferation of relevant diagrams and the increasing dimensionality of theirassociated integrals; in any case, this task can be reasonably carried on only in theframework of an automated environment.On the road towards the creation of such an environment, we here show how a strategyclosely inspired by that of Nickel and co-workers can be stated in algorithmic form, andsuccessfully implemented on a computer. As an application, we plot the minimizeddistributions of residual integrations for the sets of diagrams needed to obtain RG functionsto the full seven-loop level; they represent a good evaluation of the computational effortwhich will be required to improve the currently available estimates of critical exponents.

A general creation-annihilation model with absorbing states

A one dimensional non-equilibrium stochastic model is proposed where each site of the lattice is occupied by a particle, which may be of type A or B. The time evolution of the model occurs through three processes: autocatalytic generation of A and B particles and spontaneous conversion A ® B. The two-parameter phase diagram of the model is obtained in one- and two-site mean field approximations, as well as through numerical simulations and exact solution of finite systems extrapolated to the thermodynamic limit. A continuous line of transitions between an active and an absorbing phase is found. This critical line starts at a point where the model is equivalent to the contact process and ends at a point which corresponds to the voter model, where two absorbing states coexist. Thus, the critical line ends at a point where the transition is discontinuous. Estimates of critical exponents are obtained through the simulations and finite-size-scaling extrapolations, and the crossover between universality classes as the voter model transition is approached is studied.

Collapsing lattice animals and lattice trees in two dimensions

We present high statistics simulations of weighted lattice bond animals and lattice trees onthe square lattice, with fugacities for each non-bonded contact and for each bond betweentwo neighbouring monomers. The simulations are performed using a newly developedsequential sampling method with resampling, very similar to the pruned-enrichedRosenbluth method (PERM) used for linear chain polymers. We determine with highprecision the line of second-order transitions from an extended to a collapsed phase inthe resulting two-dimensional phase diagram. This line includes critical bondpercolation as a multicritical point, and we verify that this point divides the line intodifferent universality classes. One of them corresponds to the collapse driven bycontacts and includes the collapse of (weakly embeddable) trees. There is someevidence that the other is subdivided again into two parts with different universalityclasses. One of these (at the far side from collapsing trees) is bond driven and isrepresented by the Derrida–Herrmann model of animals having bonds only (nocontacts). Between the critical percolation point and this bond-driven collapseseems to be an intermediate regime, whose other end point is a multicritical pointP* where atransition line between two collapsed phases (one bond driven and the other contact driven) sparks off. Thispoint P* seems to be attractive (in the renormalization group sense) from the side of theintermediate regime, so there are four universality classes on the transition line(collapsing trees, critical percolation, intermediate regime, and Derrida–Herrmann). Weobtain very precise estimates for all critical exponents for collapsing trees. It isalready harder to estimate the critical exponents for the intermediate regime.Finally, it is very difficult to obtain with our method good estimates of the criticalparameters of the Derrida–Herrmann universality class. As regards the bond-drivento contact-driven transition in the collapsed phase, we have some evidence forits existence and rough location, but no precise estimates of critical exponents.