Anderson transition in 1D systems with spatial disorder

A simple Kronig-Penney model for 1D mesoscopic systems with δ peak potentials is used to study numerically the influence of spatial disorder on conductance fluctuations and distribution at different regimes. The Lévy laws are used to investigate the statistical properties of the eigenstates. It is found that an Anderson transition occurs even in 1D meaning that the disorder can also provide constructive quantum interferences. The critical disorder W_c for this transition is estimated. In these 1D systems, the metallic phase is well characterized by a Gaussian conductance distribution. Indeed, the results relative to conductance distribution are in good agreement with the previous works in 2D and 3D systems for other models. At this transition, the conductance probability distribution has a system size independent shape with large fluctuations in good agreement with previous works.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

Quantum critical behavior of the cluster glass phase

In disordered itinerant magnets with arbitrary symmetry of the order parameter, the conventional quantum critical point between the ordered phase and the paramagnetic Fermi liquid (PMFL) is destroyed due to the formation of an intervening cluster glass (CG) phase. In this Letter, we discuss the quantum critical behavior at the CG-PMFL transition for systems with continuous symmetry. We show that fluctuations due to quantum Griffiths anomalies induce a first-order transition from the PMFL at T = 0, while at higher temperatures a conventional continuous transition is restored. This behavior is a generic consequence of enhanced non-Ohmic dissipation caused by a broad distribution of energy scales within any quantum Griffiths phase in itinerant systems.     

Reconciling conductance fluctuations and the scaling theory of localization

We reconcile the phenomenon of mesoscopic conductance fluctuations with the single parameter scaling theory of the Anderson transition. We calculate three averages of the conductance distribution, exp(), , and 1/, where g is the conductance in units of e(2)/h and R = 1/g is the resistance, and demonstrate that these quantities obey single parameter scaling laws. We obtain consistent estimates of the critical exponent from the scaling of all these quantities.     

Efficient Bayesian inference for learning in the Ising linear perceptron and signal detection in CDMA

Efficient new Bayesian inference technique is employed for studying critical properties of the Ising linear perceptron and for signal detection in code division multiple access (CDMA). The approach is based on a recently introduced message passing technique for densely connected systems. Here we study both critical and non-critical regimes. Results obtained in the non-critical regime give rise to a highly efficient signal detection algorithm in the context of CDMA; while in the critical regime one observes a first-order transition line that ends in a continuous phase transition point. Finite size effects are also studied.     

Nonlinear transport near a quantum phase transition in two dimensions

The problem of nonlinear transport near a quantum phase transition is solved within the Landau theory for the dissipative insulator-superconductor phase transition in two dimensions. Using the nonequilibrium Schwinger round-trip Green function formalism, we obtain the scaling function for the nonlinear conductivity in the quantum-disordered regime. We find that the conductivity scales as E2 at low fields but crosses over at large fields to a universal constant on the order of e(2)/h. The crossover between these two regimes obtains when the length scale for the quantum fluctuations becomes comparable to that of the electric field within logarithmic accuracy.     

Kondo effect and mesoscopic fluctuations

Two important themes in nanoscale physics in the last two decades are correlations between electrons and mesoscopic fluctuations. Here we review our recent work on the intersection of these two themes. The setting is the Kondo effect, a paradigmatic example of correlated electron physics, in a nanoscale system with mesoscopic fluctuations; in particular, we consider a small quantum dot coupled to a finite reservoir (which itself may be a large quantum dot). We discuss three aspects of this problem. First, in the high-temperature regime, we argue that a Kondo temperature T _k which takes into account the mesoscopic fluctuations is a relevant concept: for instance, physical properties are universal functions of T/T _k. Secondly, when the temperature is much less than the mean level spacing due to confinement, we characterize a natural cross-over from weak to strong coupling. This strong coupling regime is itself characterized by well-defined single-particle levels, as one can see from a Nozières Fermi-liquid theory argument. Finally, using a mean-field technique, we connect the mesoscopic fluctuations of the quasiparticles in the weak coupling regime to those at strong coupling.     

Effects of counterion fluctuations in a polyelectrolyte brush

We investigate the effect of counterion fluctuations in a single polyelectrolyte brush in the absence of added salt by systematically expanding the counterion free energy about Poisson-Boltzmann mean-field theory. We find that for strongly charged brushes, there is a collapse regime in which the brush height decreases with increasing charge on the polyelectrolyte chains. The transition to this collapsed regime is similar to the liquid-gas transition, which has a first-order line terminating at a critical point. We find that, for monovalent counterions, the transition is discontinuous in theta solvent, while for multivalent counterions, the transition is generally continuous. For collapsed brushes, the brush height is not independent of grafting density as it is for osmotic brushes, but scales linear with it.Received: 26 November 2003, Published online: 11 May 2004PACS: 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling - 61.20.Qg Structure of associated liquids: electrolytes, molten salts, etc.     

Scaling of fluctuations and critical exponents

We present a new technique to describe the abnormal behavior of certain fluctuation observables in the critical regime of quantum statistical systems which undergo a phase transition. The idea is to rescale the local fluctuation operators by a relevant external parameter of the system, in addition to the usual scaling with the inverse square root of the volume. The scaling indices used in this scaling procedure are directly related to the critical exponents. Furthermore, it is explained that this new method of scaling preserves the CCR structure of the algebra of macroscopic fluctuations. Finally, scaling indices are computed for the relevant microscopic observables at all temperatures in a mean field approximation for a quantum anharmonic crystal. These indices yield the same critical exponents as predicted by mean field theory.     

Pattern formation during deformation of a confined viscoelastic layer: from a viscous liquid to a soft elastic solid

We study pattern formation during tensile deformation of confined viscoelastic layers. The use of a model system [poly(dimethylsiloxane) with different degrees of cross-linking] allows us to go continuously from a viscous liquid to an elastic solid. We observe two distinct regimes of fingering instabilities: a regime called "elastic" with interfacial crack propagation, where the fingering wavelength scales only with the film thickness, and a bulk regime called "viscoelastic," where the fingering instability shows a Saffman-Taylor-like behavior. We find good quantitative agreement with theory in both cases and present a reduced parameter describing the transition between the two regimes and allowing us to predict the observed patterns over the whole range of viscoelastic properties.     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Absence of an abrupt phase change from polycrystalline to amorphous in silicon with deposition temperature.

Using fluctuation electron microscopy, we have observed an increase in the mesoscopic spatial fluctuations in the diffracted intensity from vapor-deposited silicon thin films as a function of substrate temperature from the amorphous to polycrystalline regimes. We interpret this increase as an increase in paracrystalline medium-range order in the sample. A paracrystal consists of topologically crystalline grains in a disordered matrix; in this model the increase in ordering is caused by an increase in the grain size or density. Our observations are counter to the previous belief that the amorphous to polycrystalline transition is a discontinuous disorder-order phase transition.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.Dedicated to Ilya Prigogine on the occasion of his 70th birthday.     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

Transitions between flow regimes of baroclinic flow in a rotating annulus of fluid

Baroclinic flow in a rotating annulus of fluid shows remarkable transitions of flow patterns as do Rayleigh–Benard convection and Taylor vortices. There are four flow regimes in two nondimensional parameter space, called a symmetric regime (Hadley regime), a steady wave regime (Rossby regime), a vacillating wave regime and a geostrophic turbulence regime. Laminar flow in a symmetric regime is formed between the balance of a horizontal pressure gradient force and a Coriolis torque (geostrophic balance), and this flow becomes unstable when one of the nondimensional parameters, the thermal Rossby number, becomes less than the critical value. In this paper, the characteristic features of the four flow regimes are reviewed including recent findings about the behavior of geostrophic turbulence.     

Material yielding and irreversible deformation mediated by dislocation motion

We study the collective behavior of dislocation assemblies in simplified models of plastic deformation. We first review several numerical results on long range dislocation interactions with simplified dislocation motion constraints. These typically give rise to a yielding transition separating stationary and moving dislocation phases. Furthermore, we discuss the intermittent relaxation of the plastic strain-rate observed around this transition at mesoscopic scales, and how this intermittent behavior gives rise to an average slow power law relaxation in time known in the literature as Andrade’s creep. We analyze the coherent dynamics and the average stress-strain relationship in the steady regime of plastic deformation. In this steady regime, plastic deformation proceeds in the form of plastic avalanches whose size and duration are broadly distributed and statistically characterized. One signature of the time correlations of this heterogeneous collective dislocation dynamics is a power spectrum scaling with frequency as f ^?a with an exponent α close to 1.5. This feature appears to be peculiar of dislocation and grain boundary motion as has been observed in other physical situations in the vicinity of a yielding transition.     

Moderate point: Balanced entropy and enthalpy contributions in soft matter

Various soft materials share some common features, such as significant entropic effect, large fluctuations, sensitivity to thermodynamic conditions, and mesoscopic characteristic spatial and temporal scales. However, no quantitative definitions have yet been provided for soft matter, and the intrinsic mechanisms leading to their common features are unclear. In this work, from the viewpoint of statistical mechanics, we show that soft matter works in the vicinity of a specific thermodynamic state named moderate point, at which entropy and enthalpy contributions among substates along a certain order parameter are well balanced or have a minimal difference. Around the moderate point, the order parameter fluctuation,the associated response function, and the spatial correlation length maximize, which explains the large fluctuation, the sensitivity to thermodynamic conditions, and mesoscopic spatial and temporal scales of soft matter, respectively. Possible applications to switching chemical bonds or allosteric biomachines determining their best working temperatures are also briefly discussed.     

Temperature dependence of the gap size near the brillouin-zone nodes of HgBa2CuO4+delta superconductors

Although more than 20 years have passed, the identification of the superconducting order parameter in cuprates is still under debate. Here, we show that the gap size near the nodes is a good candidate for the order parameter: it scales with the critical temperature Tc over a wide doping range and displays a significant temperature dependence below Tc in both the underdoped and the overdoped regimes. In contrast, the gap size at the antinodes does not scale with Tc in the underdoped regime and appears to be controlled by the pseudogap which persists below Tc.     

Multi-Scale Jacobi Method for Anderson Localization

We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green’s function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.