Correlation Length Critical Exponent Research Articles

First order transition in two dimensional coulomb glass

We have studied the ground states of two dimensional lattice model of Coulomb Glass via Monte Carlo annealing. Our results show a possibility of existence of a critical disorder (Wc) below which the system is in the charge ordered phase and above it the system is in the disordered phase. We have used finite size scaling to calculate Wc = 0.2413, the critical exponent of magnetization β = 0 indicating discontinuity in magnetization and the critical exponent of correlation length ν = 1.0. The distribution of staggered magnetization for different disorder strengths shows a three peak structure. We thus predict that two dimensional Coulomb Glass shows a first order transition at T=0.

Critical properties of the quantum rotor model with phase frustration at incommensurate filling in two dimensions

The critical properties of the zero-temperature superfluid-insulator transition in the fullyfrustrated quantum rotor model at incommensurate filling in two dimensions are studied. We develop a spherical approximation scheme for the model with phase frustration f = 1/2 by choosing a primitive unit cell composed of two sites. The finite-size scaling behavior of various physical quantities in the spherical limit provides the correlation length critical exponent ν = 0.5 and the dynamical critical exponent z = 2, which are consistent with the scenario of the generic superfluidinsulator transition even in the presence of phase frustration.

From discrete to continuous percolation in dimensions 3 to 7

We propose a method of studying the continuous percolation of aligned objects as a limit of a corresponding discrete model. We show that the convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent . This allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions with accuracy far better than that attained using any other method before. We also report improved values of the correlation length critical exponent ν in dimensions d = 4,5 and the values of several universal wrapping probabilities for .

Continuum percolation of congruent overlapping spherocylinders.

Continuum percolation of randomly orientated congruent overlapping spherocylinders (composed of cylinder of height H with semispheres of diameter D at the ends) with aspect ratio α=H/D in [0,∞) is studied. The percolation threshold ϕ_{c}, percolation transition width Δ, and correlation-length critical exponent ν for spherocylinders with α in [0, 200] are determined with a high degree of accuracy via extensive finite-size scaling analysis. A generalized excluded-volume approximation for percolation threshold with an exponent explicitly depending on both aspect ratio and excluded volume for arbitrary α values in [0,∞) is proposed and shown to yield accurate predictions of ϕ_{c} for an extremely wide range of α in [0, 2000] based on available numerical and experimental data. We find ϕ_{c} is a universal monotonic decreasing function of α and is independent of the effective particle size. Our study has implications in percolation theory for nonspherical particles and composite material design.

Interface localization near criticality

The theory of interface localization in near-critical planar systems at phase coexistence is formulated from first principles. We show that mutual delocalization of two interfaces, amounting to interfacial wetting, occurs when the bulk correlation length critical exponent $\nu$ is larger than or equal to 1. Interaction with a boundary or defect line involves an additional scale and a dependence of the localization strength on the distance from criticality. The implications are particularly rich in the boundary case, where delocalization proceeds through different renormalization patterns sharing the feature that the boundary field becomes irrelevant in the delocalized regime. The boundary delocalization (wetting) transition is shown to be continuous, with surface specific heat and layer thickness exponents which can take values that we determine.

Superconductor-insulator transition of Josephson-junction arrays on a honeycomb lattice in a magnetic field

We study the superconductor to insulator transition at zero temperature in a Josephson-junction array model on a honeycomb lattice with $f$ flux quantum per plaquette. The path integral representation of the model corresponds to a (2+1)-dimensional classical model, which is used to investigate the critical behavior by extensive Monte Carlo simulations on large system sizes. In contrast to the model on a square lattice, the transition is found to be first order for $f=1/3$ and continuous for $f=1/2$ but in a different universality class. The correlation-length critical exponent is estimated from finite-size scaling of vortex correlations. The estimated universal conductivity at the transition is approximately four times its value for $f=0$. The results are compared with experimental observations on ultrathin superconducting films with a triangular lattice of nanoholes in a transverse magnetic field.

Zeros of Partition Function and Critical Exponents of the 3D Diluted Ising Model

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

Kibble-Zurek dynamics in an array of coupled binary Bose condensates

Universal dynamics of spontaneous symmetry breaking is central to understanding the universal behavior of spontaneous defect formation in various systems from the early universe, condensed-matter systems to ultracold atomic systems. We explore the universal real-time dynamics in an array of coupled binary atomic Bose-Einstein condensates in optical lattices, which undergo a spontaneous symmetry breaking from the symmetric Rabi oscillation to the broken-symmetry self-trapping. In addition to Goldstone modes, there exist gapped Higgs modes whose excitation gap vanishes at the critical point. In the slow passage through the critical point, we analytically find that the symmetry-breaking dynamics obeys the Kibble-Zurek mechanism. From the scalings of bifurcation delay and domain formation, we numerically extract two Kibble-Zurek exponents, and , which give the static correlation-length critical exponent ν and the dynamic critical exponent z. Our approach provides an efficient way for the simultaneous determination of the critical exponents ν and z for a continuous phase transition.

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

We studied the large-scale properties of directed lattice animals and directed percolation on a square lattice. Using a transfer-matrix approach on strips of finite widths, we generated relatively long sequences of estimates for effective values of critical fugacity, percolation threshold and correlation length critical exponents. We applied two different extrapolation methods to obtain estimates for infinite systems. The precision of our final estimates is comparable to (or better than) the precision of the best currently available results.

Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model

We use a simple real-space renormalization-group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model in which the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization-group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.

Reservoir interactions during Bose-Einstein condensation: Modified critical scaling in the Kibble-Zurek mechanism of defect formation

As a test of the Kibble-Zurek mechanism (KZM) of defect formation, we simulate the Bose-Einstein condensation transition in a toroidally confined Bose gas using the stochastic projected Gross-Pitaevskii equation (SPGPE), with and without the energy-damping reservoir interaction. Energy-damping alters the scaling of the winding number distribution with the quench time - a departure from the universal KZM theory that relies on equilibrium critical exponents. Numerical values are obtained for the correlation-length critical exponent $\nu$ and the dynamical critical exponent $z$ for each variant of reservoir interaction theory. The energy-damping reservoir interactions cause significant modification of the dynamical critical exponent of the phase transition, whilst preserving the essential KZM critical scaling behavior. Comparison of numerical and analytical two-point correlation functions further illustrates the effect of energy damping on the correlation length during freeze out.

The nature of the continuous non-equilibrium phase transition of Axelrod's model

Axelrod's model in the square lattice with nearest-neighbors interactions exhibits culturally homogeneous as well as culturally fragmented absorbing configurations. In the case in which the agents are characterized by F = 2 cultural features and each feature assumes k states drawn from a Poisson distribution of parameter q, these regimes are separated by a continuous transition at . Using Monte Carlo simulations and finite-size scaling we show that the mean density of cultural domains μ is an order parameter of the model that vanishes as with at the critical point. In addition, for the correlation length critical exponent we find and for Fisher's exponent, . This set of critical exponents places the continuous phase transition of Axelrod's model apart from the known universality classes of non-equilibrium lattice models.

Critical behavior of the absorbing transition of branching and annihilating bosonic random walkers

In this work, we investigate the critical behavior of a model describing the parity-conserving branching and annihilating process of random walkers. The model is simulated on a one dimensional lattice on which the sites can be occupied by multiple particles with a finite annhilation probability. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and spacial correlation length critical exponents. Further, we follow the short-time critical relaxation to provide a set of relevant dynamical critical exponents. We check the validity of the hyperscaling relation for both sets of stationary and dynamical critical exponents. These are consistent with the BARW-PC universality class.

Critical exponents ofO(N)models in fractional dimensions

We compute critical exponents of O(N) models in fractal dimensions between two and four, and for continuos values of the number of field components N, in this way completing the RG classification of universality classes for these models. In d=2 the N-dependence of the correlation length critical exponent gives us the last piece of information needed to establish a RG derivation of the Mermin-Wagner theorem. We also report critical exponents for multi-critical universality classes in the cases N>1 and N=0. Finally, in the large-N limit our critical exponents correctly approach those of the spherical model, allowing us to set N~100 as threshold for the quantitative validity of leading order large-N estimates.

The effect of quenched bond disorder on first-order phase transitions

We investigate the effect of quenched bond disorder on the two-dimensional three-color Ashkin–Teller model, which undergoes a first-order phase transition in the absence of impurities. This is one of the simplest and striking models in which quantitative numerical simulations can be carried out to investigate emergent criticality due to disorder rounding of first-order transition. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to 128×128 spins, each of which is represented by three colors taking values ±1, we show that the rounding of the first-order phase transition is an emergent criticality. We further calculate the correlation length critical exponent, ν, and the magnetization critical exponent, β, from finite size scaling analysis. We find that the critical exponents, ν and β, change as the strength of disorder or the four-spin coupling varies, and we show that the critical exponents appear not to be in the Ising universality class. We know of no analytical approaches that can explain our non-perturbative results. However our results should inspire further work on this important problem, either numerical or analytical.

Superfluid-insulator transition of the two-dimensional fully-frustrated quantum rotor model in the spherical limit

We study the critical properties of the superfluid-insulator transition in the fully-frustrated quantum rotor model at commensurate filling in two dimensions. By developing a large-N method that is applicable to the model with frustration, we find the finite-size scaling properties in the spherical limit, from which the correlation length critical exponent ? = 1.0 and the dynamical critical exponent z = 1 are confirmed. A universal sheet resistance of Rc/R0 = 1.25(1) at the critical point, where R0 is the quantum resistance, is obtained, which is consistent with the theoretical prediction.

Deconfined criticality for the two-dimensional quantum S = 1-spin model with the three-spin and biquadratic interactions

The criticality between the nematic and valence-bond-solid (VBS) phases was investigated for the two-dimensional quantum S = 1-spin model with the three-spin and biquadratic interactions by means of the numerical diagonalization method. It is expected that the criticality belongs to a novel universality class, the so-called deconfined criticality, accompanied with unconventional critical indices. In this paper, we incorporate the three-spin interaction, and adjust the (redundant) interaction parameter so as to optimize the finite-size behavior. Treating the finite-size cluster with N ≤ 20 spins, we estimate the correlation-length critical exponent as ν = 0.88(3).

Study of the antiferromagnetic Blume-Capel model by using the partition function zeros in the complex temperature plane

The critical temperature and the thermal scaling exponent of the square-lattice Blume-Capel model are obtained as functions of fugacity by calculating the partition function zeros in the complex temperature plane of the square-lattice Blume-Capel antiferromagnet for the first time. The thermal scaling exponent yt determines the specific-heat critical exponent and the correlation-length critical exponent. The value of yt = 2 indicates a strong first-order phase transition in two dimensions whereas yt = 1 corresponds to an Ising-class second-order phase transition. The full spectrum (from yt = 2 to yt = 1) of the thermal scaling exponent for the square-lattice Blume-Capel model is evaluated for the first time. The phase diagram of the square-lattice Blume-Capel model, including the tricritical point, is also obtained, and it is compared with other results in the literature.

Universality and the QCD Anderson Transition

We study the Anderson-type transition previously found in the spectrum of the QCD quark Dirac operator in the high-temperature, quark-gluon plasma phase. Using finite size scaling for the unfolded level spacing distribution, we show that in the thermodynamic limit there is a genuine mobility edge, where the spectral statistics changes from Poisson to Wigner-Dyson statistics in a nonanalytic way. We determine the correlation length critical exponent ν and find that it is compatible with that of the unitary Anderson model.

From characterisation of pore-structures to simulations of pore-scale fluid flow and the upscaling of permeability using microtomography: A case study of heterogeneous carbonates

We propose a workflow to provide detailed characterisation of microstructures, simulations of permeability at micro-scale, and to analyse the upscaling of permeability. The workflow is fully tested using two very different microtomographic datasets of carbonate samples with strong heterogeneity. The characterisation of microstructures includes not only routine parameters but also anisotropy of the pore-structure. The critical pore diameter is also obtained through morphological implementations. The method of stochastic analysis of extended local porosity theory is used to determine the sizes of representative volume elements (RVE). We demonstrate criteria for determining the size of the RVE and show a sample that satisfies the criterion. We also discuss a sample that does not suffice the RVE criterion, because of the strong anisotropy. Permeabilities are computed using Lattice-Boltzmann (LB) simulations on the RVE and compared with laboratory measurements. Techniques and procedures are used to extract scaling parameters for both of the samples including: 1) percolation threshold — by using a shrinking/expanding algorithm; 2) crossover length — by analysing the mass density; 3) the critical exponent of correlation length — by using the finite-size scaling scheme; and 4) the critical exponent of permeability — by running LB simulations on a series of derivative models close to the percolation threshold. Our results of critical exponents are different from the previous studies and the mass density distributions are irregular. This pilot study provides new information on the relationships between microstructures and permeability of natural rocks with complex microstructures. The study also reveals the scaling parameters. Our results clearly put into question whether natural rocks can be idealised by classical theoretical solutions. A robust workflow for embracing a computational approach of microstructures may provide a solution.