Two-dimensional Random Systems Research Articles

Self-organized criticality of magnetic avalanches in disordered ferrimagnetic material.

We observe multiple steplike jumps in a Dy-Fe-Ga-based ferrimagnetic alloy in its magnetic hysteresis curve at 2K. The observed jumps are found to have a stochastic character with respect to their magnitude and the field position, and the jumps do not correlate with the duration of the field. The distribution of jump size follows a power law variation indicating the scale invariance nature of the jumps. We have invoked a simple two-dimensional random bond Ising-type spin system to model the dynamics. Our computational model can qualitatively reproduce the jumps and their scale-invariant character. It also elucidates that the flipping of antiferromagnetically coupled Dy and Fe clusters is responsible for the observed jumps in the hysteresis loop. These features are described in terms of the self-organized criticality.

Controllable random lasers based on two-dimensional random gain systems with nematic liquid crystals

Controllable random lasers based on two-dimensional (2D) random gain systems with nematic liquid crystals are investigated by using the finite-difference time-domain (FDTD) methods. The dependence of random lasers on the scattering strength of nematic liquid crystals is studied. The results show that the random laser characteristics including the pump threshold, the emission intensity and the number of laser modes can be controlled by varying the pump rates, the filling fraction of nematic liquid crystals, and the included angle between the polarization direction of pump beam and the director of nematic liquid crystals. To achieve the optimal random lasers, the polarization direction of pump beam should be parallel to the director of nematic liquid crystals.

Transport of velocity alignment particles in random obstacles**Project supported by the National Natural Science Foundation of China (Grant Nos. 11575064 and 11175067), the PCSIRT (Grant No. IRT1243), the GDUPS (2016), and the Natural Science Foundation of Guangdong Province, China (Grant No. 2014A030313426).

We numerically investigate the trapping behaviors of aligning particles in two-dimensional (2D) random obstacles system. Under the circumstances of the effective diffusion rate and the average velocity tend to zero, particles are in trapped state. In this paper, we examine how the system parameters affect the trapping behaviors. At the large self-propelled speed, the ability of nematic particles escape from trapping state is enhancing rapidly, in the meanwhile the polar and free particles are still in trapped state. For the small rotation diffusion coefficient, the polar particles circle around (like vortices) the obstacles and here particles are in trapped state. Interestingly, only the partial nematic particles are trapped in the confined direction and additional particles remain flowing. In the free case, the disorder particle–particle collisions impede the motion in each other’s directions, leading the free particles to be trapped. At the large rotation diffusion coefficient, the ordered motion of aligning particles disappear, particles fill the sample evenly and are self-trapped around obstacles. As the particles approach the trapping density due to the crowding effect the particles become so dense that they impede each other’s motion. With the increasing number of obstacles, the trajectories of particles are blocked by obstacles, which obstruct the movement of particles. It is worth noting that when the number of the obstacles are large enough, once the particles are trapped, the system is permanently absorbed into a trapped state.

Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses

Abstract This work explores the percolation thresholds of continuum systems consisting of randomly-oriented overlapping ellipses. High-precision percolation thresholds for various homogeneous ellipse systems with different aspect ratios are obtained from extensive Monte Carlo simulations based on the incorporation of Vieillard-Baron’s contact function of two identical ellipses with our efficient algorithm for continuum percolation. In addition, we generalize Vieillard-Baron’s contact function from identical ellipses to unequal ellipses, and extend the Monte Carlo algorithm to heterogeneous ellipse systems where the ellipses have different dimensions and/or aspect ratios. Based on the concept of modified excluded area, a general law is verified for precise prediction of percolation threshold for many heterogeneous ellipse systems. In particular, the study of heterogeneous ellipse systems gains insight into the apparent percolation threshold symmetry observed earlier in systems comprising unequal circles (Consiglio et al., 2004).

Rayleigh approximation to ground state of the Bose and Coulomb glasses.

Glasses are rigid systems in which competing interactions prevent simultaneous minimization of local energies. This leads to frustration and highly degenerate ground states the nature and properties of which are still far from being thoroughly understood. We report an analytical approach based on the method of functional equations that allows us to construct the Rayleigh approximation to the ground state of a two-dimensional (2D) random Coulomb system with logarithmic interactions. We realize a model for 2D Coulomb glass as a cylindrical type II superconductor containing randomly located columnar defects (CD) which trap superconducting vortices induced by applied magnetic field. Our findings break ground for analytical studies of glassy systems, marking an important step towards understanding their properties.

Thermalization of Lévy Flights: Path-Wise Picture in 2D

We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which in the presence of confining potentials may asymptotically set down at Boltzmann-type thermal equilibria. In view of the Eliazar-Klafter no-go statement, such dynamical behavior is plainly incompatible with the standard Langevin modeling of Lévy flights. No explicit path-wise description has been so far devised for the thermally equilibrating random motion we address, and its formulation is the principal goal of the present work. To this end we prescribe a priori the target pdf ρ∗ in the Boltzmann form ~exp[] and next select the Lévy noise (e.g., its Lévy measure) of interest. To reconstruct random paths of the underlying stochastic process we resort to numerical methods. We create a suitably modified version of the time honored Gillespie algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation.

From percolating to dense random stick networks: Conductivity model investigation

In a Monte Carlo study the conductivity of two-dimensional random stick systems is investigated from the percolation threshold up to ten times the percolation threshold density. We propose a model explicitly depending on the stick density and junction-to-stick conductance ratio. The model describes the transition from the conductivity determined by the structure of a percolating cluster to the conductivity of the dense random stick networks. The model is motivated by the observed densities of the sticks and contacts involved in the current flow. The finite-size scaling effects are also included in the description. The derived model for conductivity should be broadly applicable to the random networks of the rodlike particles.

Quantum diffusion in two-dimensional random systems with particle–hole symmetry

We study the scattering dynamics of an n-component spinor wavefunction in a random environment on a two-dimensional lattice. If the particle–hole symmetry of the Hamiltonian is spontaneously broken the dynamics of the quantum particles becomes diffusive on large scales. The latter is described by a non-interacting Grassmann field, indicating a special kind of asymptotic freedom on large scales in d = 2.

Metal-insulator transition in two-dimensional random fermion systems of chiral symmetry classes

Field-theoretical approach to Anderson localization in 2D disordered fermionic systems of chiral symmetry classes (BDI, AIII, CII) is developed. Important representatives of these symmetry classes are random hopping models on bipartite lattices at the band center. As was found by Gade and Wegner two decades ago within the sigma-model formalism, quantum interference effects in these classes are absent to all orders of perturbation theory. We demonstrate that the quantum localization effects emerge when the theory is treated non-perturbatively. Specifically, they are controlled by topological vortex-like excitations of the sigma models. We derive renormalization group equations including these non-perturbative contributions. Analyzing them, we find that the 2D disordered systems of chiral classes undergo a metal-insulator transition driven by topologically induced Anderson localization. We also show that the Wess-Zumino and Z_2 theta terms on surfaces of 3D topological insulators (in classes AIII and CII, respectively) overpower the vortex-induced localization.

Light localization characteristics in a random configuration of dielectric cylindrical columns

In the two-dimensional random system composed of a disordered array of dielectric cylindrical columns, Anderson localization of light occurs. To obtain frequency dependence of the light localization characteristics, we have simulated temporal diffusion of electromagnetic waves in such a random system adopting parameters of actual nanosized semiconductor samples with a high filling fraction of the columns, using the finite-difference time-domain method. We have investigated diffusion length, autocorrelation function of light energy density, and time variation in total energy within the system at several frequencies. We obtain universal behavior of light localization phenomenon as a function of the light localization length and system size, from which we estimate frequency dependence of the localization length. In addition, we show that the frequency dependence of the localization effect depends on the degree of wave interference due to Bragg-like diffraction, rather than on the magnitude of the light scattering cross section of a single scatterer.

Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization

Quasimodes of an open finite-size two-dimensional (2D) random system are computed and systematically characterized in terms of their spatial extension $\ensuremath{\eta}$, complexity factor ${q}^{2}$, and phase distribution for a collection of random systems ranging from weakly scattering to localized systems. A rapid change is seen in $\ensuremath{\eta}$ and ${q}^{2}$ at the crossover from localized to diffusive which corresponds to the emergence of 2D extended multipeaked quasimodes analogous to the necklace states recently observed in one dimension. These 2D quasimodes are interpreted in terms of coupled localized states.

Dynamics of wave packets in two-dimensional random systems with anisotropic disorder

A theoretical model is proposed to describe narrowband pulse dynamics in two-dimensional systems with arbitrary correlated disorder. In anisotropic systems with elongated cigarlike inhomogeneities, fast propagation is predicted in the direction across the structure where the wave is exponentially localized and tunneling of evanescent modes plays a dominant role in typical realizations. Along the structure, where the wave is channeled as in a waveguide, the motion of the wave energy is relatively slow. Numerical simulations performed for ultra-wide-band pulses show that even at the initial stage of wave evolution, the radiation diffuses predominantly in the direction along the major axis of the correlation ellipse. Spectral analysis of the results relates the long tail of the wave observed in the transverse direction to a number of frequency domain "lucky shots" associated with the long-living resonant modes localized inside the sample.

Universal distribution of spectral-flow gaps in the Rashba model with disorder

We report a study of disordered electron systems with spin-orbit coupling on a cylinder using methods of random matrix ensembles. With a threading flux turned on, the single-particle levels will generally avoid, rather than cross, each other. Our numerical study of the level-avoiding gaps in the disordered Rashba model demonstrates that the normalized gap distribution is of a universal form, independent of the random strength and the system size. For small gaps, it exhibits a linear behavior, while for large gaps, it decays exponentially. A framework based on matrix mechanical models is suggested and is verified to reproduce the universal linear behavior at small gaps. Thus, we propose the use of the distribution of the spectral-flow gaps associated with flux insertion as a way to characterize two-dimensional random systems with spin-orbit coupling. The relevance and qualitative implications for spin (Hall) transport are also addressed.

Coherent electron dynamics in a two-dimensional random system with mobilityedges

We study numerically the dynamics of a one-electron wavepacket in a two-dimensionalrandom lattice with long-range correlated diagonal disorder in the presence of a uniformelectric field. The time-dependent Schrödinger equation is used for this purpose. We findthat the wavepacket displays Bloch-like oscillations associated with the appearance of aphase of delocalized states in the strong correlation regime. The amplitude ofoscillations directly reflects the bandwidth of the phase and allows us to measure it.The oscillations reveal two main frequencies whose values are determined by thestructure of the underlying potential in the vicinity of the wavepacket maximum.

Two-dimensional driven diffusion in a random potential

In one dimension, the emergence of non-equilibrium net currents is characteristic for periodically driven systems with broken spatial-reversal symmetry. We show that a similar effect also exists in two-dimensional random systems due to inevitable local symmetry breaking in the potential landscape. We study the properties of the steady-state net currents that emerge in the problem of driven diffusive motion in a random uncorrelated potential under alternating square-wave external field bias. The net current distribution function is found to be approximated well by a Gaussian and the average magnitude of the net currents is defined by higher than linear order even terms in the response function. Furthermore, the net currents can form highly correlated patterns, with large-scale vorticity emerging in the system, in the regime of small and intermediate alternating field amplitudes. With an increase in the amplitude, the vorticity patterns undergo a gradual randomization, approaching a random configuration limit for large values of the external bias. The correlation length decreases as a function of the applied field amplitude and grows nearly linearly with the system size.

Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems.

We rigorously calculate the propagation and scattering of electromagnetic waves by rectangular and random arrays of dielectric cylinders in a uniform medium. For regular arrays, the band structures are computed and complete bandgaps are discovered. For random arrays, the phenomenon of wave transmission and scattering is investigated and compared in two scenarios: (1) Wave propagating through the array of cylinders; this is the scenario which has been commonly considered in the literature, and (2) wave transmitted from a source located inside the ensemble. We show that within complete band gaps, results from the two scenarios are similar. Outside the gaps, however, there could be a distinct difference, that is, wave transmission can be inhibited by disorders in the first scenario, but such an inhibition may not prevail in the second scenario.

Simulation study of localization of electromagnetic waves in two-dimensional random dipolar systems.

We study the propagation and scattering of electromagnetic waves by random arrays of dipolar cylinders in a uniform medium. A set of self-consistent equations, incorporating all orders of multiple scattering of the electromagnetic waves, is derived from first principles and then solved numerically for electromagnetic fields. For certain ranges of frequencies, spatially localized electromagnetic waves appear in such a simple but realistic disordered system. Dependence of localization on the frequency, radiation damping, and filling factor is shown. The spatial behavior of the total, coherent, and diffusive waves is explored in detail, and found to comply with a physical intuitive picture. A phase diagram characterizing localization is presented, in agreement with previous investigations on other systems.

Diffusive and localization behavior of electromagnetic waves in a two-dimensional random medium.

In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan et al. [J. Opt. Soc. Am. B 10, 391 (1993)]. A set of self-consistent equations is presented, accounting for the multiple scattering in the system, and is then solved numerically. A strong localization regime is discovered in the frequency domain. The transport properties within, near the edge of, and nearly outside the localization regime are investigated for different parameters such as filling factor and system size. The results show that within the localization regime, waves are trapped near the transmitting source. Meanwhile, the diffusive waves follow an intuitive but expected picture. That is, they increase with traveling path as more and more random scattering incurs, followed by a saturation, then start to decay exponentially when the travelling path is large enough, signifying the localization effect. For the cases where the frequencies are near the boundary of or outside the localization regime, the results of diffusive waves are compared with the diffusion approximation, showing less encouraging agreement as in other systems [Asatryan et al., Phys. Rev. E 67, 036605 (2003)].

Magnetic properties of two-dimensional random systems of Ising dipoles

Within the framework of the generalized mean-field theory, two-dimensional random systems of point Ising magnetic dipoles of both fluid and lattice type are considered. In the former case, the precise expression for the distribution function of random magnetic fields is found, the impossibility of ferromagnetic ordering established and the susceptibility of the system in a paramagnetic state is calculated. In the latter case, it has been shown that the ground state of the system of Ising dipoles placed in randomly filled sites of two-dimensional square lattice, depends on the fraction p of the filled sites: at p> p c=0.63, the system is ferromagnetic, otherwise—paramagnetic. The transition between those states is of percolation nature. Temperature dependencies of the magnetization in the ferromagnetic phase and susceptibility of the system in the paramagnetic phase have been found.

Chaotic transition of random dynamical systems and chaos synchronization by common noises

We studied the mechanism behind the connection between the transition to chaos of random dynamical systems and the synchronization of chaotic maps driven by external common noises. Near the chaotic transition, the spatial size of random dynamical systems shows an extreme intermittent behavior. By calculating the scaling exponents, we have found that the origin of this intermittent behavior is on-off intermittency. This led us to conclude that chaotic transitions through on-off intermittency can be regarded as a route for random dynamical systems. To clarify this argument, a two-dimensional random dynamical system and two coupled logistic maps driven by external common noises were analyzed.