Diverging Length Scale Research Articles

NONEQUILIBRIUM MULTICRITICAL BEHAVIOR IN ANISOTROPIC HEISENBERG FERROMAGNET DRIVEN BY OSCILLATING MAGNETIC FIELD

The Heisenberg ferromagnet (uniaxially anisotropic along z-direction), in the presence of time dependent (but uniform over space) magnetic field, is studied by Monte Carlo simulation. The time dependent magnetic field was taken as elliptically polarised in such a way that the resulting field vector rotates in the XZ-plane. In the limit of low anisotropy, the dynamical responses of the system are studied as functions of temperature and the amplitudes of the magnetic field. As the temperature decreases, it was found that the system undergoes multiple dynamical phase transitions. In this limit, the multiple transitions were studied in details and the phase diagram for this observed multicritical behavior was drawn in field amplitude and temperature plane. The natures (continuous/discontinuous) of the transitions are determined by the temperature variations of fourth-order Binder cumulant ratio and the distributions of order parameters near the transition points. The transitions are supported by finite size study. The temperature variations of the variances of dynamic order parameter components (for different system sizes) indicate the existence of diverging length scale near the dynamic transition points. The frequency dependences of the transition temperatures of the multiple dynamic transition are also studied briefly.

Localization Transition of the Three-Dimensional Lorentz Model and Continuum Percolation

The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation theory and obtain an excellent matching of the critical density and exponents. Within a dynamic scaling ansatz incorporating two divergent length scales we achieve data collapse for the mean-square displacements and identify the leading corrections to scaling. We provide evidence for a divergent non-Gaussian parameter close to the transition.

Dynamics and thermodynamics of the glass transition

The principal theme of this paper is that anomalously slow, super-Arrhenius relaxations in glassy materials may be activated processes involving chains of molecular displacements. As pointed out in a preceding paper with Lemaitre, the entropy of critically long excitation chains can enable them to grow without bound, thus activating stable thermal fluctuations in the local density or molecular coordination of the material. I argue here that the intrinsic molecular-scale disorder in a glass plays an essential role in determining the activation rate for such chains, and show that a simple disorder-related correction to the earlier theory recovers the Vogel-Fulcher law in three dimensions. A key feature of this theory is that the spatial extent of critically long excitation chains diverges at the Vogel-Fulcher temperature. I speculate that this diverging length scale implies that, as the temperature decreases, increasingly large regions of the system become frozen and do not contribute to the configurational entropy, and thus ergodicity is partially broken in the super-Arrhenius region above the Kauzmann temperature T(K). This partially broken ergodicity seems to explain the vanishing entropy at T(K) and other observed relations between dynamics and thermodynamics at the glass transition.

1∕dexpansion fork-core percolation

The physics of k-core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para H2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k = 1) and biconnected (k = 2) percolation, the mean field k > or = 3-core percolation transition is both continuous and discontinuous, i.e., there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1/d expansion for k-core percolation on the d-dimensional hypercubic lattice. We show that to order 1/d3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field k-core transition survives in high dimensions.

Vibrations and Diverging Length Scales Near the Unjamming Transition

We numerically study the vibrations of jammed packings of particles interacting with finite-range, repulsive potentials at zero temperature. As the packing fraction phi is lowered towards the onset of unjamming at phi(c), the density of vibrational states approaches a nonzero value in the limit of zero frequency. For phi >phi(c), there is a crossover frequency, omega* below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with phi-phi(c). Characteristic length scales, determined from the dominant wave vector contributing to the eigenmode at omega*, diverge as power laws at the unjamming transition.

Nonequilibrium criticality at shock formation in steady states

The steady state shock formation in processes like nonconserving asymmetric simple exclusion processes in varied situations is shown to be a nonequilibrium critical phenomenon. The diverging length scales and the quantitative description of the transition including the phase boundary are obtained from a few general properties of the dynamics without relying on specific details.

Barkhausen Noise, Subloops, Demagnetization Curves, and Universal Scaling Tuning Disorder, Field Sweep Rate, and History

AbstractFor Abstract see ChemInform Abstract in Full Text.

Fluid adsorption near an apex: covariance between complete and critical wetting.

Critical wetting is an elusive phenomenon for solid-fluid interfaces. Using interfacial models we show that the diverging length scales, which characterize complete wetting at an apex, precisely mimic critical wetting with the apex angle behaving as the contact angle. Transfer matrix, renormalization group, and mean-field analysis show that this covariance is obeyed in 2D and 3D and for long- and short-ranged forces. This connection should be experimentally accessible and provides a means of checking theoretical predictions for critical wetting.

Peculiar scaling of self-avoiding walk contacts.

The nearest neighbor contacts between the two halves of an N-site lattice self-avoiding walk offer an unusual example of scaling random geometry: for N-->infinity they are strictly finite in number but their radius of gyration R(c) is power law distributed proportional to R(-tau)(c), where tau>1 is a novel exponent characterizing universal behavior. A continuum of diverging length scales is associated with the R(c) distribution. A possibly superuniversal tau = 2 is also expected for the contacts of a self-avoiding or random walk with a confining wall.

Equilibrium crystal shapes in the Potts model.

The three-dimensional q-state Potts model, forced into coexistence by fixing the density of one state, is studied for q=2, 3, 4, and 6. As a function of temperature and number of states, we studied the resulting equilibrium droplet shapes. A theoretical discussion is given of the interface properties at large values of q. We found a roughening transition for each of the numbers of states we studied, at temperatures that decrease with increasing q, but increase when measured as a fraction of the melting temperature. We also found equilibrium shapes closely approaching a sphere near the melting point, even though the three-dimensional Potts model with three or more states does not have a phase transition with a diverging length scale at the melting point.

Subloops, Barkhausen noise, and disorder induced critical behavior

Hysteresis loops are often seen in experiments at first order phase transformations when the system goes out of equilibrium, such as in the supercooling of liquids and in magnets. The nonequilibrium, zero-temperature random-field Ising model has been studied as a model for the hysteretic behavior of these transformations. As disorder is added, one finds a transition where the jump in the saturation hysteresis loop (corresponding to an infinite avalanche) decreases to zero. At this transition the model exhibits power law distributions of noise (avalanches), universal behavior, and a diverging length scale [O. Perković, K. Dahmen, and J. P. Sethna, Phys. Rev. B 59, 6106 (1999)]. Interestingly, not only the saturation loops but also subloops reflect this critical point, and at the critical disorder one finds history-induced critical scaling. We present simulation results for histories in systems of almost 14 million spins. Concentric inner subloops are found to resemble rescaled saturation loops at effectively higher (possibly correlated) disorder. In addition, avalanche size distributions for the inner subloops are collapsed using Widom scaling methods. The resulting exponents and scaling functions are shown to differ from those corresponding to the saturation loop.

Hysteresis and avalanches: phase transitions and critical phenomena in driven disordered systems

We discuss Barkhausen noise in magnetic systems in terms of avalanches near a disorder-induced critical point, using the hysteretic zero-temperature random-field Ising model and recent variants. As the disorder is decreased, one finds a transition from smooth hysteresis loops to loops with a sharp jump in magnetization (corresponding to an infinite avalanche). In a large region near the transition point the model exhibits power-law distributions of noise (avalanches), universal behavior and a diverging length scale. Universal properties of this critical point are reported that were obtained using renormalization group methods and numerical simulations. Connections to other experimental systems such as athermal martensitic phase transitions (with and without ‘bursts’) and front propagation are also discussed.

Solitons in the One-Dimensional Forest Fire Model

Fires in the one-dimensional Bak-Chen-Tang forest fire model propagate as solitons, resembling shocks in Burgers turbulence. The branching of solitons, creating new fires, is balanced by the pairwise annihilation of oppositely moving solitons. Two distinct, diverging length scales appear in the limit where the growth rate of trees, p, vanishes. The width of the solitons, w, diverges as a power law, 1/p, while the average distance between solitons diverges much faster as d approximately exp(pi2/12p).

Bond-disordered Anderson model on a two-dimensional square lattice: Chiral symmetry and restoration of one-parameter scaling

The bond-disordered Anderson model in two dimensions on a square lattice is studied numerically near the band center by calculating the density of states (DOS), multifractal properties of eigenstates, and the localization length. The DOS divergence at the band center is studied and compared with Gade's result and power law. Although Gade's form describes accurately the DOS of finite-size systems near the band center, it fails to describe the calculated part of the DOS of the infinite system, and a different expression is proposed. Study of the level spacing distributions reveals that the state closest to the band center and the next one have a different level spacing distribution than pairs of states away from the band center. Multifractal properties of finite systems furthermore show that the scaling of eigenstates changes discontinuously near the band center. This unusual behavior suggests the existence of a divergent length scale, whose existence is explained as the finite-size manifestation of the band center critical point of the infinite system, and the critical exponent of the correlation length is calculated by a finite-size scaling. Furthermore, study of the scaling of the Lyapunov exponents of transfer matrices of long stripes indicates that for a long stripe of any width there is an energy region around the band center within which the Lyapunov exponents cannot be described by one-parameter scaling. This region vanishes, however, in the limit of the infinite square lattice, when one-parameter scaling is restored, and the scaling exponent calculated is in agreement with the result of the finite-size scaling analysis.

Avalanches, Barkhausen noise, and disorder-induced critical behavior

Hysteresis loops are often seen in experiments at first-order phase transformations when the system goes out of equilibrium. They may have a macroscopic jump, roughly as seen in the supercooling of liquids, or they may be smoothly varying, as seen in most magnets. We have studied the nonequilibrium zero-temperature random-field Ising model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition the model exhibits power law distributions of noise (avalanches), universal behavior, and a diverging length scale, which should be detectable in acoustic emission and other noise measurements. We study universal properties of this critical point using renormalization group methods and numerical simulations. Connections to experimental systems such as athermal martensitic phase transitions (with and without ‘‘bursts’’) and the Barkhausen effect in magnetic systems will be discussed. Similar ideas can also be applied to the interpretation of the Gutenberg–Richter scaling law in the statistics of earthquakes.

Hysteresis, Barkhausen noise, and disorder induced critical behavior

Hysteresis loops are often observed in experiments in first order phase transformations when the system goes out of equilibrium. They may have a macroscopic jump, roughly as seen in the supercooling of liquids, or they may be smoothly varying, as seen in most magnets. The nonequilibrium zero-temperature random-field Ising-model can be used to model hysteretic behavior in first order phase transformations: as the disorder is decreased, one finds a transition from smooth hysteresis loops to loops with a sharp jump in magnetization (corresponding to an infinite avalanche). In a large region near the transition point the model exhibits power law distributions of noise (avalanches), universal behavior and a diverging length scale. Universal properties of this critical point are reported that were obtained using renormalization group methods and numerical simulations. Connections to experimental systems such as athermal martensitic phase transitions (with and without bursts) and the Barkhausen effect in magnetic systems are discussed.

Properties of spin-triplet, even-parity superconductors

The physical consequences of the spin-triplet, even-parity pairing that has been predicted to exist in disordered two-dimensional electron systems are considered in detail. We show that the presence of an attractive interaction in the particle-particle spin-triplet channel leads to an instability of the normal metal that competes with the localizing effects of the disorder. The instability is characterized by a diverging length scale, and has all of the characteristics of a continuous phase transition. The transition and the properties of the ordered phase are studied in mean-field theory, and by taking into account Gaussian fluctuations. We find that the ordered phase is indeed a superconductor with an ordinary Meissner effect and a free energy that is lower than that of the normal metal. Various technical points that have given rise to confusion in connection with this and other manifestations of odd-gap superconductivity are also discussed.

Multi-scaling properties of phase segregation in quenched binary fluids: evidence from simulations

The phase separation process in two-dimensional binary fluid systems is investigated using molecular dynamics for almost 20 000 particles. Previous works from the same authors have shown that the late-stage coarsening regime at critical volume fractions is described by power laws whose exponents are dependent on particle–particle interactions and on temperature in a non-universal way. At the low-temperature region, however, the emergence of a three-phase regime of percolating domains with atoms of type- A, type- B and voids, invalidates the single-length scale invariance. We argue in terms of a multiple-length scale analysis. In particular, for the case we studied with symmetric molecular pair potentials we propose two divergent length scales to characterize the dynamic process: one associated to the average size of equal-atom domains and the other related to the average size of undistinguishable-atom domains. Two different growth exponents can be inferred in such a case but the system does not exhibit scale invariance.

Why Two Renormalization Groups are Better Than One

The advantages of using more than one renormalization group (RG) in problems with more than one important length scale are discussed. It is shown that: i) using different RG's can lead to complementary information, i.e. what is very difficult to calculate with an RG based on one flow parameter may be much more accesible using another; ii) using more than one RG requires less physical input in order to describe via RG methods the theory as a function of its parameters; iii) using more than one RG allows one to describe problems with more than one diverging length scale. The above points are illustrated concretely in the context of both particle physics and statistical physics using the techniques of environmentally friendly renormalization. Specifically, finite temperature λφ4 theory, an Ising-type system in a film geometry, an Ising-type system in a transverse magnetic field, the QCD coupling constant at finite temperature and the crossover between bulk and surface critical behavior in a semi-infinite system are considered.

Properties of the random-field Ising model in a transverse magnetic field

We consider the effect of a random longitudinal field on the Ising model in a transverse magnetic field. For spatial dimension $d > 2$, there is at low strength of randomness and transverse field, a phase with true long range order which is destroyed at higher values of the randomness or transverse field. The properties of the quantum phase transition at zero temperature are controlled by a fixed point with no quantum fluctuations. This fixed point also controls the classical finite temperature phase transition in this model. Many critical properties of the quantum transition are therefore identical to those of the classical transition. In particular, we argue that the dynamical scaling is activated, i.e, the logarithm of the diverging time scale rises as a power of the diverging length scale.