Random‐field ising systems: A survey of current theoretical views

Ising or Ising-like models in random fields are good representations of a large number of impure materials. The main attempts of theoretical treatments of these models--as far as they are relevant from an experimental point of view--are reviewed. A domain argument invented by Imry and Ma shows that the long-range order is not destroyed by weak random-fields in more than D = 2 dimensions. This result is supported by considerations of the roughening of an isolated domain wall in such systems: domain walls turn out to be well defined objects for D > 2, but arbitrarily convoluted for D < 2. Different approaches for calculating the roughness exponent ζ yield ζ= (5 - D)/3 in random-field systems. The application of ζ in incommensurate-commensurate critical behaviour is discussed.

Non-classical critical behaviour occurs in random-field systems below D = 6 dimensions which is determined in general by three independent exponents. The new exponent y_J = θ= D/2 - σ corresponds to random-field renormalization or, to say it differently, to the irrelevance of the temperature at the zero-temperature fixed point, which is believed to rule the critical behaviour. The inequalities satisfied by these exponents are investigated, as well as the relations between the eigenvalue and the critical exponents and their numerical values found in the literature.

Domain wail roughening due to random fields produces also metastability and hysteresis. It is shown that when cooling a 3D system into the low-temperature phase in an applied random field, the system runs into a metastable domain state, in agreement with the experimental observation. The further approach of the system to the ordered equilibrium state is hindered by pinning which leads to domain size increasing only logarithmically in time. Domain wall roughness and pinning in random bond systems is also considered.     

Random and aperiodic quantum spin chains: A comparative study

According to the Harris-Luck criterion the relevance of a fluctuating interaction at the critical point is connected to the value of the fluctuation exponent . Here we consider different types of relevant fluctuations in the quantum Ising chain and investigate the universality class of random as well as deterministic-aperiodic models. At the critical point the random and aperiodic systems behave similarly, due to the same type of extreme broad distribution of the energy scales at low energies. The critical exponents of some averaged quantities are found to be a universal function of , but some others do depend on other parameters of the distribution of the couplings. In the off-critical region there is an important difference between the two systems: there are no Griffiths singularities in aperiodic models. Received: 18 November 1997 / Received in final form: 24 November 1997 / Accepted: 8 January 1997     

Observational constraints of modified Chaplygin gas in loop quantum cosmology

We explore the behavior of the holographic superconductors at zero temperature for a charged scalar field coupled to a Maxwell field in higher-dimensional AdS soliton spacetime via analytical way. In the probe limit, we obtain the critical chemical potentials increase linearly as a total dimension d grows up. We find that the critical exponent for condensation operator is obtained as 1/2 independently of d, and the charge density is linearly related to the chemical potential near the critical point. Furthermore, we consider a slightly generalized setup the Einstein–Power–Maxwell field theory, and find that the critical exponent for condensation operator is given as 1/(4?2n) in terms of a power parameter n of the Power–Maxwell field, and the charge density is proportional to the chemical potential to the power of 1/(2?n).     

Griffiths-McCoy singularities in random quantum spin chains: exact results through renormalization

The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.     

Localization properties of two interacting particles in a quasi-periodic potential with a metal-insulator transition

We study the influence of many-particle interactions on a metal-insulator transition. We consider the two-interacting-particle problem for onsite interacting particles on a one-dimensional quasiperiodic chain, the so-called Aubry-André model. We show numerically by the decimation method and finite-size scaling that the interaction does not modify the critical parameters such as the transition point and the localization-length exponent. We compare our results to the case of finite density systems studied by means of the density-matrix renormalization scheme. Received 28 June 2001     

Percolation in two-scale porous media

Two-scale porous media are generated by filtering a Gaussian random correlated field with a random correlated threshold field. The percolation threshold and the critical exponent ν are derived with the help of a finite-size scaling method. The percolation threshold for the three-dimensional media is a decreasing function of the variance and correlation length of the threshold field. A simplified model predicts these trends in 3d; moreover, it suggested some effects in 2d which were all numerically verified. Received 17 August 2000     

三维随机点阵Ising模型的集团Monte Carlo方法模拟

为了解决Monte Carlo模拟中的临界慢化问题,应用S-W方法,对一维随机点阵的模型进行了研究。找到该系统的相变点为β_c=0.075±0.001,与此同时,在相变点附近研究了SW方法的动力学性质,计算得动力学临界指数z=0.74±0.03,结果表明,SW方法对克服三维随机点阵Ising模型的临界慢化是非常有效的。 关键词：     

Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models

We find the crossover behaviour of the disorder averaged spin-spin correlation function for the 2D Ising and 3-state Potts model with random bonds at the critical point. The procedure employed is the renormalisation approach of the perturbation series around the conformal field theories representing the pure models. We obtain a crossover in the amplitude for the correlation function for the Ising model, which does not change the critical exponent, and a shift in the critical exponent produced by randomness in the case of the Potts model. A comparison with numerical data is discussed briefly.     

Electric permittivity anomaly close to the critical mixing point: An improved method of estimation of the exponent describing a weak critical anomaly

A critical anomaly of electric permittivity in the vicinity of the consolute critical point of an ethanol and dodecane mixture has been measured and analyzed. A method of analysing the permittivity anomaly, which enhances the credibility of the fitting, has been proposed. We have found out that the exponent predicted in theoretical expectations, 1-α = 0.89, describes the anomaly correctly. However, in spite of the application of the improved method, it is not possible to obtain the value of the fitted critical exponent with satisfying precision.     

Disease extinction in the presence of random vaccination

We investigate disease extinction in an epidemic model described by a birth-death process. We show that, in the absence of vaccination, the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point, with an unusual critical exponent. Even a comparatively weak Poisson-distributed random vaccination leads to an exponential increase in the extinction rate, with the exponent that strongly depends on the vaccination parameters.     

On the universality of geometrical and transport exponents of rigidity percolation

We develop a three-parameter position-space renormalization group method and investigate the universality of geometrical and transport exponents of rigidity (vector) percolation in two dimensions. To do this, we study site-bond percolation in which sites and bonds are randomly and independently occupied with probabilitiess andb, respectively. The global flow diagram of the renormalization transformation is obtained which shows that thegeometrical exponents of the rigid clusters in both site and bond percolation belong to the same universality class, and possibly that of random (scalar) percolation. However, if we use the same renormalization transformation to calculate the critical exponents of the elastic moduli of the system in bond and site percolation, we find them to be very different (although the corresponding values of the correlation length exponent are the same). This indicates that the critical exponent of the elastic moduli of rigidity percolation may not be universal, which is consistent with some of the recent numerical simulations.     

Critical phenomena in a disc-percolation model and their application to relativistic heavy ion collisions

By studying the critical phenomena in continuum-percolation of discs, we find a new approach to locate the critical point, i.e.using the inflection point of P_∞ as an evaluation of the percolation threshold.The susceptibility, defined as the derivative of P_∞, possesses a finite-size scaling property, where the scaling exponent is the reciprocal of ν, the critical exponent of the correlation length.A possible application of this approach to the study of the critical phenomena in relativistic heavy ion collisions is discussed.The critical point for deconfinement can be extracted by the inflection point of P_(QGP)-the probability for the event with QGP formation.The finite-size scaling of its derivative can give the critical exponent ν, which is a rare case that can provide an experimental measure of a critical exponent in heavy ion collisions.     

Black-hole singularities: a new critical phenomenon

The singularity inside a spherical charged black hole, coupled to a spherical, massless scalar field is studied numerically. The profile of the characteristic scalar field was taken to be a power of advanced time with an exponent alpha>0. A critical exponent alpha(crit) exists. For exponents below the critical one (alphaalpha(crit)) an all-encompassing, spacelike singularity evolves, which completely blocks the "tunnel" inside the black hole, preventing the use of the black hole as a portal for hyperspace travel.     

Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.     

Critical behaviour of random compressible magnets

The critical properties of a compressible random magnet are studied using renormalization group methods. Then-component orderparameter is coupled to quenched disorder and to the elastic fluctuations of the anisotropic solid. It is shown, that the critical behaviour of a compressible random magnet is in general the same as that of a random magnet on a rigid lattice. However, if the specific heat exponent of the ideal magnet is positive and the disorder is sufficiently weak, a macroscopic instability may prevent the system in reaching the critical point. The resulting first-order transition may be preceded by pseudocritical behaviour characteristic to pure compressible magnets. The effect of random magnetic fields on the critical properties of compressible magnets is also discussed.     

Quantum phase transition of alkaline-earth fermionic atoms confined in an optical superlattice

Using the density matrix renormalization group method, we evaluate the spin and charge gaps of alkaline-earth fermionic atoms in a periodic one-dimensional optical superlattice. The number of delocalized atoms is equal to the lattice size and we consider an antiferromagnetic coupling between delocalized and localized atoms. We found a quantum phase transition from a Kondo insulator spin liquid state without confining potential to a charge-gapped antiferromagnetic state with nonzero potential. For each on-site coupling, there is a critical potential point for which the spin gap vanishes and its value increases linearly with the local interaction.     

Exotic versus conventional scaling and universality in a disordered bilayer quantum heisenberg antiferromagnet

We present Monte Carlo simulations of a two-dimensional bilayer quantum Heisenberg antiferromagnet with random dimer dilution. In contrast with exotic scaling scenarios found in other random quantum systems, the quantum phase transition in this system is characterized by a finite-disorder fixed point with power-law scaling. After accounting for corrections to scaling, with a leading irrelevant exponent of omega approximately 0.48, we find universal critical exponents z=1.310(6) and nu=1.16(3). We discuss the consequences of these findings and suggest new experiments.     

Critical collapse of a complex scalar field with angular momentum

We report a new critical solution found at the threshold of axisymmetric gravitational collapse of a complex scalar field with angular momentum. To carry angular momentum the scalar field cannot be axisymmetric; however, its azimuthal dependence is defined so that the resulting stress-energy tensor and spacetime metric are axisymmetric. The critical solution found is nonspherical, discretely self-similar with an echoing exponent Delta=0.42(+/-4%), and exhibits a scaling exponent gamma=0.11(+/-10%) in near-critical collapse. Our simulations suggest that the solution is universal (within the imposed symmetry class), modulo a family-dependent constant, complex phase.     

Gedanken experiments at high-order approximation: nearly extremal Reissner-Nordström black holes cannot be overcharged

Phenomenologically interesting scalar potentials are highly atypical in generic random landscapes. We develop the mathematical techniques to generate constrained random potentials, i.e. Slepian models, which can globally represent low-probability realizations of the landscape. We give analytical as well as numerical methods to construct these Slepian models for constrained realizations of a full Gaussian random field around critical as well as inflection points. We use these techniques to numerically generate in an efficient way a large number of minima at arbitrary heights of the potential and calculate their non-perturbative decay rate. Furthermore, we also illustrate how to use these methods by obtaining statistical information about the distribution of observables in an inflationary inflection point constructed within these models.     

Metastability and Spinodal Points for a Random Walker on a Triangle

We investigate time-dependent properties of a single-particle model in which a random walker moves on a triangle and is subjected to nonfocal boundary conditions. This model exhibits spontaneous breaking of a Z ₂ symmetry. The reduced size of the configuration space (compared to related many-particle models that also show spontaneous symmetry breaking) allows us to study the spectrum of the time evolution operator. We break the symmetry explicitly and find a stable phase, and a metastable phase which vanishes at a spinodal point. At this point, the spectrum of the time evolution operator has a gapless and universal band of excitations with a dynamical critical exponent z=1. Surprisingly, the imaginary parts of the eigenvalues E _j(L) are equally spaced, following the rule . Away from the spinodal point, we find two time scales in the spectrum. These results are related to scaling functions for the mean path of the random walker and to first passage times. For the spinodal point, we find universal scaling behavior. A simplified version of the model which can be handled analytically is also presented.