Estimates Of Critical Exponents Research Articles

Accurate estimates of 3D Ising critical exponents using the coherent-anomaly method

An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by α = 0.108(5), β = 0.327(4), γ = 1.237(4) and δ = 4.77(5).

Critical phenomena in delayed maps

A phase transition mediated by chaotic defect dynamics is observed in a simple discrete-time model of a delayed dynamical system. A numerical estimate of critical exponents permits one to insert the transition in a proper universality class, within percolation phenomena.

Chain polymers near an adsorbing surface

We present improved estimates of critical exponents for 3D polymers near a plane adsorbing surface. The improvements were possible by simulating rather long chains (up to N=2000) by means of a new algorithm. Apart from minor but significant adjustments in the location of the tricritical point and in the various gamma -exponents, our main result is that the crossover exponent for critical adsorption is compatible with phi = 1/2 . This is much lower than previous estimates and suggests that this exponent might be superuniversal.

EXTREME TYPE II SUPERCONDUCTORS: UNIVERSAL PROPERTIES AND TRENDS

We review and analyze experiment results of the specific heat, the muon–spin rotation (μSR) relaxation rate and the fluctuation contributions to the magnetization and dc conductivity of extreme type II superconductors from the point of view of critical phenomena. Our estimates of critical exponents and amplitudes, the measured scaling behavior, the consistencies with the universal relations between the critical amplitudes and the decrease of the transition temperature (T c ) with decreasing thickness, which corresponds to a dimensional crossover from 3d and 2d xy critical behavior, provide considerable evidence for a three-dimensional xy critical point. The estimated volume of the critical correlation length amplitudes turns out to be comparable to that in super-fluid Helium and is several orders of magnitude smaller than in BCS superconductors. Moreover, motivated by the Uemura plot, which relates the measured T c to the zero temperature μSR relaxation rate, and by the hyperuniversal relation between T c and critical amplitudes of the London penetration depth and phase correlation length, we propose a simple "universal" scaling ansatz, where the plot of the rescaled transition temperature versus rescaled μSR relaxation rate should fall on a single parabola. Our analysis of the μSR data reveals excellent agreement with this scaling ansatz for a large class of cuprate and Chevrel-phase superconductors. The resulting dependence of T c on the zero-temperature "condensate density" is then used to explore universal trends in the pressure (α) and isotope (β) coefficients. In good agreement with experiment we find that α and β fall into common T c -α and T c -β regions, respectively, forming two branches, one for systems with positive and the other for compounds with negative pressure or isotope coefficient. The two branches merge at the maximum T c where the coefficients vanish and the magnitude of the coefficients increases with decreasing T c . The "universal" scaling ansatz also implies that the critical amplitudes of the penetration depth and phase correlation length are related to the zero temperature condensate density. This quantity is also related to the hole concentration. Finally we discuss the temperature dependence of the penetration depth. μSR measurements indicate that for various cuprates the temperature dependence for 0 < T < T c appears to be bounded by the two-fluid model and the dilute charged Bose gas behaviors, respectively. Consistent with the dependence of T c on the zero temperature condensate density, compounds with high T c 's turn out to be closer to two-fluid behavior, while compounds with decreasing T c and in turn with lower condensate density clearly reveal the crossover to the dilute gas limit. These trends combined with T → 0 point uniquely to Bose condensation of interacting and weakly charged pairs as the mechanism that drives the transition.

Critical behaviour of a surface reaction model with infinitely many absorbing states

In a recent letter, Yaldram et al.(1993) studied the critical behaviour of a simple lattice gas model of the CO-NO catalytic reaction. The model exhibits a second-order non-equilibrium phase transition from an active state into one out of infinitely many absorbing states. Estimates for the critical exponent beta suggested that the model belongs to a new universality class. The results reported in this article contradict this notion, as estimates for various critical exponents show that the model belongs to the universality class of directed percolation.

Computer simulation of critical phenomena in fluids

The author extends to the computer simulation of off-lattice models a phenomenological description of finite-size effects, already successfully used in the simulation of lattice systems. The density fluctuations are studied in subsystems of the simulation cell in the canonical ensemble. The density distribution functions of the subsystems are analysed by using finite-size scaling. Results are presented for the two-dimensional Lennard-Jones fluid with N=4096 particles. The study of the reduced fourth-order cumulants of the density distribution functions allows him to obtain the critical temperature and density. He compares the results with previous values obtained by different methods. He shows that finite-size scaling concepts can be extended to off-lattice models. A much larger computer effort would be necessary, however, in order to get reliable estimates of critical exponents and amplitudes.

The Crumpling Transition of Dynamically Triangulated Random Surfaces

We present the crumpling transition in three-dimensional Euclidian space of dynamically triangulated random surfaces with edge extrinsic curvature and fixed topology of a sphere as well as simulations of a dynamically triangulated torus. We used longer runs than previous simulations and give new and more accurate estimates of critical exponents. Our data indicate a cusp singularity in the specific heat. The transition temperature, as well as the exponents are topology dependent.

Extreme type II superconductors

Abstract We shall review and analyze experimental results of the specific heat, chemical potential, muon-spin (μSR)-relaxation rate and fluctuation contributions to the magnetization and DC conductivity of extreme type II superconductors in the context of critical phenomena. Our estimates of critical exponents and amplitudes, together with measured scaling behavior and the consistency with the universal relations between the critical amplitudes, provide considerable evidence for a three-dimensional xy-critical point. The associated order parameter is a complex scalar describing singlet and s-wave pairing. The estimated volume of the critical correlation length amplitudes turns out to be comparable to that in superfluid helium and is several orders of magnitude smaller than in BCS superconductors. Moreover, motivated by the Uemura plot, which relates the measured transition temperature (Tc) to the zero-temperature μSR-relaxation rate, and by the hyperuniversal relation between Tc and critical amplitudes of the London penetration depth and phase correlation length, we propose a simple ansatz, where the rescaled transition temperature and μSR-relaxation rate should fall on a single parabola. Our analysis of the μSR data reveals excellent agreement with this scaling ansatz for a large class of cuprate and Chevrel-phase superconductors. The resulting dependence of Tc on the zero-temperature condensate density is then used to explore universal trends in the pressure (α) and isotope (β) coefficients. In good agreement with experiment we find that α and β fall into common Tc-α and Tc-β regions, respectively, thus forming two branches, one for systems with positive and the other for compounds with negative pressure or isotope coefficient. The two branches merge at the maximum Tc where the coefficients vanish and the magnitude of the coefficients increases with reduced Tc. Our results provide crucial constraints on the microscopic theory.

Quantum percolation on a face-centered-cubic lattice.

We report numerical results for the quantum-percolation problem on a face-centered-cubic lattice. The quantum-percolation threshold, mobility edges, and localization lengths are calculated by studying the scaling behavior of the sensitivity of eigenvalues to a change in boundary conditions. We give estimates of critical exponents for the conductivity and the localization length near the mobility edge. We find it difficult to interpret our results in the framework of a one-parameter scaling theory. The influence of the asymmetric density of states and the presence of odd-membered rings is discussed.

Low-dimensional quantum antiferromagnets: Criticality and series expansions at zero temperature

The theory of scaling for critical points occuring at zero temperature is described. One new exponent, z;- y, is required: this is positive if T is a relevant variable but negative if, as in the random-field Ising model for d>2, a transition for T>0 is controlled by a zero-temperature fixed point. Recent series extrapolation studies by Gelfand, Singh and Huse on the phase transitions and ordering properties of one- and two-dimensional quantal antiferromagnets at zero temperature are reviewed [1–9]. Power series in various parameters about exact Ising and about singlet dimerized ground states have been generated by cluster expansion techniques and quantum perturbation theory carried to orders 5 to 15. Results for phases, phase diagrams, and estimates for critical exponents and other parameters are consistent with analytic analyses [10–12] and experiments [13] and reveal significant new features.

New numerical method to study phase transitions.

A powerful method of detecting first order transitions by numerical simulations of finite systems is presented. The method relies on simulations and the finite size scaling properties of free energy barriers between coexisting states. It is demonstrated that the first order transitions in d = 2, q = 5 and d = q = 3 Potts models are easily seen with modest computing time. The method can also be used to obtain quite accurate estimates of critical exponents by studying the barriers in the vicinity of a critical point. Some new results on exponents and conformal charge in frustrated XY models and a related coupled XY-Ising model in d = 2 are presented. These show that the transitions in these models are in new universality classes and that the conformal charge varies with a parameter.

Logarithmic corrections in antiferromagnetic chains.

Various numerical estimates of critical exponents for antiferromagnetic chains are compared with conformal field-theory predictions. Field theory allows the explicit calculation of logarithmic corrections, and hence the value of the effective exponent that is expected to be observed numerically. Comparison is made with numerically obtained exponent values, and agreement is found to be very good, even when logarithmic corrections are as large as 70%. Logarithmic corrections can lead to misleading results, for example an incorrect phase diagram, and examples in the literature where this seems to have occurred are discussed.

Monte Carlo study of surface phase transitions in the three-dimensional Ising model.

We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling ${\mathit{J}}_{\mathit{s}}$. The surface-bulk multicritical point is located with improved precision, ${\mathit{J}}_{\mathit{s}}$/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}}_{1}^{\mathit{m}}$=1.41\ifmmode\pm\else\textpm\fi{}0.14, ${\ensuremath{\gamma}}_{11}^{\mathit{m}}$=0.96\ifmmode\pm\else\textpm\fi{}0.09, and along with critical amplitude ratios these are compared with theoretically predicted values.

Scaling analysis of the new multicritical behavior of CsMnBr3 and CsNiCl3.

Critical properties of the axial triangular antiferromagnet ${\mathrm{CsNiCl}}_{3}$ and the planar triangular antiferromagnet ${\mathrm{CsMnBr}}_{3}$ under an applied magnetic field are studied using scaling theory. Novel types of multicritical behavior observed in these compounds are associated with n=2 (${\mathrm{CsMnBr}}_{3}$) and n=3 (${\mathrm{CsNiCl}}_{3}$) chiral universality classes recently identified by the renormalization-group analysis. Various experimentally testable predictions are derived on the basis of a scaling ansatz combined with the numerical estimates of critical exponents. In particular, all phase boundaries emanating from the multicritical points are found to be scaled by the corresponding anisotropy-crossover exponent. It is also predicted that the criticality along the high-field critical line of ${\mathrm{CsNiCl}}_{3}$ (paramagnetic to spin flop) is of n=2 chiral universality.

Computer simulation of driven diffusive systems with exchanges

The field-driven Kawasaki model with a fractionp admixture of Glauber dynamics is studied by computer simulation:p=0 corresponds to the order-parameter-onserving driven diffusive system, whilep=1 is the equilibrium Ising model. Forp=0.1 our best estimates of critical exponents based on a system of size 4096×128 areβ≈0.22, ηRS≈0.45, andv‖≈v⊥≈1. These exponents differ from both the values predicted by a field-theoretic method forp=0 and those of the equilibrium Ising model. Anisotropic finite-size scaling analyses are carried out, both for subsystems of the large system and for fully periodic systems. The results of the latter, however, are inconsistent, probably due to the complexity of the size effects. This leaves open the possibility that we are in a crossover regime fromp=0 top≠0 and that our critical exponents are “effective ones.” Forp=0 our results are consistent with the predictionsv‖>v⊥.

Critical exponents of the dilute Ising model from four-loop expansions

Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.

Finite-size-scaling estimation of critical exponents in the two-dimensional axial next-nearest-neighbor Ising model.

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

Numerical studies of Ising spin glasses in two, three, and four dimensions.

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

CAM estimates of critical exponents of spin glasses and percolation

The coherent anomaly method (CAM) proposed by the present author is applied to spin glasses to estimate the critical exponent γ s⋍2.87 for the nonlinear susceptibility X 2 in three dimensions and the CAM theory is also demonstrated to be applicable to percolation.

Rooted spiral trees in dimensions 2,3 and 4

The authors have studied rooted spiral trees in dimensions 2, 3 and 4 by exact enumeration and by Monte Carlo methods. Numerical estimates of critical exponents thus obtained support the hypothesis that a dimensional reduction by four occurs in this problem. A possible mechanism of the origin of this dimensional reduction is proposed.