Statics and kinetics at the nematic-isotropic interface in porous media

We extend the random anisotropy nematic spin model to study nematic-isotropic transitions in porous media. A complete phase diagram is obtained. In the limit of relative low randomness the existence of a triple point is predicted. For relatively large randomness we have found a depression in temperature at the transition, together with a first order transition which ends at a tricritical point, beyond which the transition becomes continuous. We use this model to investigate the motion of the nematic-isotropic interface. We assume the system to be isothermal and initially quenched into the metastable régime of the isotropic phase. Using an appropriate form of the free energy density we obtain the domain wall solutions of the time-dependent Ginzburg-Landau equation. We find that including a random field leads to smaller velocity of the interface and to larger interface width. Received 12 November 1998 and Received in final form 15 March 1999     

Random walks on a ( 2+1)-dimensional deformable medium

A model of random walks on a deformable medium is proposed in 2+1 dimensions. The behavior of the walk is characterized by the stability parameter beta and the stiffness exponent alpha. The average square end-to-end distance l approximately equals (2nu) and the average number of visited sites approximately equals (k) are calculated. As beta increases, for each alpha there exists a critical transition point beta(c) from purely random walks ( nu = 1/2 and k approximate to 1) to compact growth ( nu = 1/3 and k = 2/3). The relationship between beta(c) and alpha can be expressed as beta(c) = e(alpha). The landscape generated by a walk is also investigated by means of the visit-number distribution N(n)(beta). There exists a scaling relationship of the form N(n)(beta)approximately n(-2)f(n/beta(z)).     

Violation of the Bell inequality in quantum critical random spin-1/2 chains

We investigate the entanglement and nonlocality properties of two random XX spin-1/2 critical chains, in order to better understand the role of breaking translational invariance to achieve nonlocal states in critical systems. We show that breaking translational invariance is a necessary but not sufficient condition for nonlocality, as the random chains remain in a local ground state up to a small degree of randomness. Furthermore, we demonstrate that the random dimer model does not have the same nonlocality properties of the translationally invariant chain, even though they share the same universality class for a certain range of randomness.     

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.     

Extended State to Localization in Random Aperiodic Chains

The electronic states in Thus-Morse chain （TMC） and generalized Fibonacci chain （GFC） are studied by solving eigenequation and using transfer matrix method. Two model Hamiltonians are studied. One contains the nearest neighbor （n.n.） hopping terms only and the other has additionally next nearest neighbor （n.n.n.） hopping terms. Based on the transfer matrix method, a criterion of transition from the extended to the localized states is suggested for CFC and TMC. The numerical calculation shows the existence of both extended and localized states in pure aperiodic system. A random potential is introduced to the diagonal term of the Hamiltonian and then the extended states are always changed to be localized. The exponents related to the localization length as a function of randomness are calculated. For different kinds of aperiodic chain, the critical value of randomness for the transition from extended to the localized states are found to be zero, consistent with the case of ordinary one-dimensional systems.     

Extended State to Localization in Random Aperiodic Chains

The electronic states in Thus-Morse chain (TMC) and generalized Fibonacci chain (GFC) are studied by solving eigenequation and using transfer matrix method. Two model Hamiltonians are studied. One contains the nearest neighbor (n.n.) hopping terms only and the other has additionally next nearest neighbor (n.n.n.) hopping terms. Based on the transfer matrix method, a criterion of transition from the extended to the localized states is suggested for GFC and TMC. The numerical calculation shows the existence of both extended and localized states in pure aperiodic system. A random potential is introduced to the diagonal term of the Hamiltonian and then the extended states are always changed to be localized. The exponents related to the localization length as a function of randomness are calculated. For different kinds of aperiodic chain, the critical value of randomness for the transition from extended to the localized states are found to be zero, consistent with the case of ordinary one-dimensional systems.     

Disorder driven melting of the vortex line lattice

The 3D XY model with random in-plane couplings is simulated to model the phase diagram of a disordered type II superconductor as a function of temperature T and randomness strength p for fixed applied magnetic field. As p increases to a critical p(c), the first order vortex lattice melting line turns parallel to the T axis, continuing down to low temperatures, rather than ending at a critical point. Above p(c) preliminary results suggest the absence of a phase coherent vortex glass.     

Non-equilibrium relaxation study of the ±J Ising model with biquadratic exchange interaction

The spin-1 ±J Ising model with uniform biquadratic couplings on a simple cubic lattice is studied by the Monte Carlo simulation using the non-equilibrium relaxation method. The reentrant phase transition induced by competition between the bilinear and biquadratic couplings is eliminated gradually with increasing randomness of bilinear couplings and disappears entirely in the strong random system. The dynamic exponent of ferromagnetic transition shows non-universal behavior with changing randomness, while this behavior is not observed in the case of staggered quadrupolar transition.     

Stability of the Haldane state against the antiferromagnetic-bond randomness

Ground-state phase diagram of the one-dimensional bond-random S=1 Heisenberg antiferromagnet is investigated by means of the loop-cluster-update quantum Monte-Carlo method. The random couplings are drawn from a rectangular uniform distribution. We found that even in the case of extremely broad bond distribution, the magnetic correlation decays exponentially, and the correlation length is hardly changed; namely, the Haldane phase continues to be realized. This result is accordant with that of the exact-diagonalization study, whereas it might contradict the conclusion of an analytic theory founded in a power-law bond distribution instead. The latter theory predicts that a second-order phase transition occurs at a certain critical randomness, and the correlation length diverges for sufficiently strong randomness. Received: 31 March 1998 / Revised and Accepted: 7 July 1998     

Social influence in small—world networks

We report on our numerical studies of the Axelrod model for social influence in small-world networks.Our simulation results show that the topology of the network has a crucial effect on the evolution of cultures .As the randomness of the network increases,the system undergoes a transition from a highly fragmented phase to a uniform phase.we also find that the power-law distribution at the transition point,reported by castellano et al,is not a critical phenomenon;it exists not only at the onset of transition but also for almost any control parameters,All these power-law distributions are stable against pertubations.A mean-field theory is developed to explain these phenomena.     

Competition between Kondo and RKKY correlations in the presence of strong randomness

We propose that competition between Kondo and magnetic correlations results in a novel universality class for heavy fermion quantum criticality in the presence of strong randomness. Starting from an Anderson lattice model with disorder, we derive an effective local field theory in the dynamical mean-field theory approximation, where randomness is introduced into both hybridization and Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions. Performing the saddle-point analysis in the U(1) slave-boson representation, we reveal its phase diagram which shows a quantum phase transition from a spin liquid state to a local Fermi liquid phase. In contrast with the clean limit case of the Anderson lattice model, the effective hybridization given by holon condensation turns out to vanish, resulting from the zero mean value of the hybridization coupling constant. However, we show that the holon density becomes finite when the variance of the hybridization is sufficiently larger than that of the RKKY coupling, giving rise to the Kondo effect. On the other hand, when the variance of the hybridization becomes smaller than that of the RKKY coupling, the Kondo effect disappears, resulting in a fully symmetric paramagnetic state, adiabatically connected to the spin liquid state of the disordered Heisenberg model. We investigate the quantum critical point beyond the mean-field approximation. Introducing quantum corrections fully self-consistently in the non-crossing approximation, we prove that the local charge susceptibility has exactly the same critical exponent as the local spin susceptibility, suggesting an enhanced symmetry at the local quantum critical point. This leads us to propose novel duality between the Kondo singlet phase and the critical local moment state beyond the Landau-Ginzburg-Wilson paradigm. The Landau-Ginzburg-Wilson forbidden duality serves the mechanism of electron fractionalization in critical impurity dynamics, where such fractionalized excitations are identified with topological excitations.     

Effects of quenched disorder in the two-dimensional Potts model: a Monte Carlo study

Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.     

Scaling of the spin stiffness in random spin- \frac{\mathsf 1}{\mathsf 2} chains

In this paper we study the localization transition induced by the disorder in random antiferromagnetic spin- chains. The results of numerical large scale computations are presented for the XX model using its free fermions representation. The scaling behavior of the spin stiffness is investigated for various disorder strengths. The disorder dependence of the localization length is studied and a comparison between numerical results and bosonization arguments is presented. A non trivial connection between localization effects and the crossover from the pure XX fixed point to the infinite randomness fixed point is pointed out.Received: 6 February 2004, Published online: 12 August 2004PACS: 75.10.Jm Quantized spin models - 75.40.Mg Numerical simulation studies - 05.70.Jk Critical point phenomena - 75.50.Lk Spin glasses and other random magnets     

Random field effects on the anisotropic quantum Heisenberg model with Gaussian random magnetic field distribution

The effect of a zero-centered Gaussian random magnetic field distribution on the phase transition properties of the anisotropic quantum Heisenberg model has been investigated on a honeycomb lattice within the framework of effective field theory (EFT) for a two-spin cluster (which is abbreviated as EFT-2). Particular attention has been devoted to investigation of the effect of the anisotropy in the exchange interaction on a system with Gaussian random magnetic field distribution. The variation of the critical temperature with the randomness parameter (i.e., the width of the distribution) has been obtained for several anisotropy parameters. Critical Gaussian distribution width values, which make the critical temperature zero, have been obtained. Moreover, it has been concluded that all critical temperatures are of second order, and that reentrant behavior does not exist in the phase diagrams.     

Critical disorder effects in Josephson-coupled quasi-one-dimensional superconductors

Effects of non-magnetic randomness on the critical temperature T _c and diamagnetism are studied in a class of quasi-one dimensional superconductors. The energy of Josephson-coupling between wires is considered to be random, which is typical for dirty organic superconductors. We show that this randomness destroys phase coherence between the wires and T _c vanishes discontinuously when the randomness reaches a critical value. The parallel and transverse components of the penetration depth are found to diverge at different critical temperatures T _c ⁽¹⁾ and T _c , which correspond to pair-breaking and phase-coherence breaking. The interplay between disorder and quantum phase fluctuations results in quantum critical behavior at T = 0, manifesting itself as a superconducting-normal metal phase transition of first-order at a critical disorder strength.     

Elastic–plastic transition in three-dimensional random materials: massively parallel simulations,fractal morphogenesis and scaling functions

Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscopically uniform loadings. The materials are cubic-shaped domains (of up to 100?×?100?×?100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J ₂ flow rule. The spatial assignment of the grains’ elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The randomness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the ‘reduced von-Mises stress’, ‘reduced plastic volume fraction’ and ‘reduced fractal dimension’), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitutive properties and their random noise levels.     

Disorder averaging and finite-size scaling

We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling predictions are fulfilled only by the new average.     

Solution Space Coupling in the Random K-Satisfiability Problem

The random K-satisfiability(K-SAT)problem is very difcult when the clause density is close to the satisfiability threshold.In this paper we study this problem from the perspective of solution space coupling.We divide a given difcult random K-SAT formula into two easy sub-formulas and let the two corresponding solution spaces to interact with each other through a coupling field x.We investigate the statistical mechanical property of this coupled system by mean field theory and computer simulations.The coupled system has an ergodicity-breaking(clustering)transition at certain critical value x d of the coupling field.At this transition point,the mean overlap value between the solutions of the two solution spaces is very close to 1.The mean energy density of the coupled system at its clustering transition point is less than the mean energy density of the original K-SAT problem at the temperature-induced clustering transition point.The implications of this work for designing new heuristic K-SAT solvers are discussed.     

Solution Space Coupling in the Random K-Satisfiability Problem

The random K-satisfiability (K-SAT) problem is very difficult when the clause density is close to the satisfiability threshold. In this paper we study this problem from the perspective of solution space coupling. We divide a given difficult random K-SAT formula into two easy sub-formulas and let the two corresponding solution spaces to interact with each other through a coupling field x. We investigate the statistical mechanical property of this coupled system by mean field theory and computer simulations. The coupled system has an ergodicity-breaking (clustering) transition at certain critical value x_d of the coupling field. At this transition point, the mean overlap value between the solutions of the two solution spaces is very close to 1. The mean energy density of the coupled system at its clustering transition point is less than the mean energy density of the original K-SAT problem at the temperature-induced clustering transition point. The implications of this work for designing new heuristic K-SAT solvers are discussed.     

Mott-Anderson transition controlled by a magnetic field in pyrochlore molybdate

The pyrochlore molybdate Gd2MO2O7 locates near the phase boundary between the ferromagnetic-metallic and the spin-glass insulating state. This metal-insulator transition is governed on a large energy scale by the electron-correlation effect, while the geometrical frustration causes the random potential. The magnetic field can tune the randomness of the potential and control, under a suitable pressure, the continuous Mott-Anderson transition precisely. The critical exponent (mu = 1.04 +/- 0.1) of the Mott-Anderson transition has been determined for this ferromagnetic orbital-degenerate electron system.