Functional renormalization description of the roughening transition

We reconsider the problem of the static thermal roughening of an elastic manifold at the critical dimension d=2 in a periodic potential, using a perturbative Functional Renormalization Group approach. Our aim is to describe the effective potential seen by the manifold below the roughening temperature on large length scales. We obtain analytically a flow equation for the potential and surface tension of the manifold, valid for low temperatures. On a length scale L, the renormalized potential is made up of a succession of quasi parabolic wells, matching onto one another in a singular region of width for large L. For strong periodic potential, the perturbation theory breaks down, and we argue, based on a variational calculation, that the transition becomes first order. We also obtain numerically the step energy as a function of temperature, and relate our results to the existing experimental data on ⁴He. Finally, we examine the case of a non local elasticity which is realized physically for the contact line. Received 16 April 1999 and Received in final form 11 October 1999     

Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

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.     

On the statistical mechanics of elastic manifolds with random interaction

We study the statistical mechanics of D-dimensional elastic manifolds, interacting via randomly distributed forces. It is shown, that this model can be mapped onto the statistical mechanics of disorder-induced roughening of a D-dimensional interface with D transverse degrees of freedom in a disordered medium. The roughness exponent ζ for the lateral deformations is calculated for different kinds of elastic response of the manifolds. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 11, 801–806 (10 December 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Continuous phase transition in a disordered eight-states Potts model

We investigate the two-dimensional eight-states ferromagnetic Potts model in the Voronoi-Delaunay tessellation. In this study, we assume that the coupling factor J varies with the distance r between the first neighbors as , with . The disordered system is simulated applying the single-cluster Monte-Carlo update algorithm and the reweighting technique. We find that this model displays a first-order phase transition if , in agreement with previous recent studies. For and 1.0, a typical second order transition is observed and the critical exponents for magnetization and susceptibility are calculated. Received 19 May 1999 and Received in final form 2 June 1999     

Comparative study of the critical behavior in one-dimensional random and aperiodic environments

We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents . At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as . Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of , whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important difference between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities. Received: 5 February 1998 / Accepted: 17 April 1998     

Finite-size scaling in the theory above the upper critical dimension

We derive exact results for several thermodynamic quantities of the O ( n ) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O ( n ) symmetric model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near T_c of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the field theory turn out to be nonuniversal whereas those of the lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions. Received: 20 October 1997 / Accepted: 5 March 1998     

Hysteresis and elastic interactions of microasperities in dry friction

Velocity independent dry friction of a slider upon a base is due to an hysteretic response of relative displacement to a tangential driving force F. We show that the purely elastic model for multistability considered in a previous publication is in no way essential: multistability arises just as well from adhesion. We emphasize the physical consequences of multistability for dynamic/static, a.c./d.c. friction. When the slider is moved from rest by an amount the transition from the zero force static configuration to dynamic behaviour is progressive, spreading on a range equal to the width of the hysteresis cycle. When is small, an elastic restoring force ensues, in agreement with observations. The competition of that elastic pinning with bulk elasticity generates a screening length which we believe is the natural size of Burridge Knopoff blocks. We then study the effect of elastic interactions between asperities: it is weak for dilute asperities, but its long range makes it important. In lowest order the interaction mediated displacement of a given asperity has logarithmically divergent fluctuations: they become comparable to the asperity radius when the slider size reaches another characteristic “Larkin length”, which for dilute micronic asperities is exponentially large. We give arguments suggesting that individually monostable asperities display collective multistability on scales larger than . For individually multistable sites we show that elastic interactions give rise to cascade processes in which the spinodal jump of a given asperity triggers the jump of others. We estimate the size of these cascades that should show up in the noise spectrum. Received: 3 February 1998 / Accepted: 19 March 1998     

Critical phenomena at perfect and non-perfect surfaces

The effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices. In particular, results of Monte Carlo simulations for flat, perfect surfaces are compared to those for flat surfaces with random, “weak” or “strong”, interactions between neighbouring spins in the surface layer, and for surfaces with steps of monoatomic height. Surface critical exponents at the ordinary transition, in particular ,are found to be robust against these perturbations. Received: 7 October 1997 / Accepted: 19 November 1997     

Universal properties of shortest paths in isotropically correlated random potentials

We consider the optimal paths in a d-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential. We study how the universal scaling exponents, the roughness and the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results using Dijkstra's algorithm to determine the optimal path in directed as well as undirected lattices indicate that the correlations become relevant if they decay with distance slower than 1/r in d = 2 and 3. We show that the exponent relation 2ν - ω = 1 holds at least in d = 2 even in case of correlations. Both in two and three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations. Received 20 December 2002 / Received in final form 10 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: schorr@lusi.uni-sb.de     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

A lattice gas coupled to two thermal reservoirs: Monte Carlo and field theoretic studies

We investigate the collective behavior of an Ising lattice gas, driven to non-equilibrium steady states by being coupled to two thermal baths. Monte Carlo methods are applied to a two-dimensional system in which one of the baths is fixed at infinite temperature. Both generic long range correlations in the disordered state and critical properties near the second order transition are measured. Anisotropic scaling, a key feature near criticality, is used to extract and some critical exponents. On the theoretical front, a continuum theory, in the spirit of Landau-Ginzburg, is presented. Being a renormalizable theory, its predictions can be computed by standard methods of -expansions and found to be consistent with simulation data. In particular, the critical behavior of this system belongs to a universality class which is quite different from the uniformly driven Ising model. Received 4 October 2000     

Model glasses coupled to two different heat baths

In a p-spin interaction spherical spin-glass model both the spins and the couplings are allowed to change with time. The spins are coupled to a heat bath with temperature T, while the coupling constants are coupled to a bath having temperature T_J. In an adiabatic limit (where relaxation time of the couplings is much larger that of the spins) we construct a generalized two-temperature thermodynamics. It involves entropies of the spins and the coupling constants. The application for spin-glass systems leads to a standard replica theory with a non-vanishing number of replicas, n=T/T _J . For p>2 there occur at low temperatures two different glassy phases, depending on the value of n. The obtained first-order transitions have positive latent heat, and positive discontinuity of the total entropy. This is an essentially non-equilibrium effect. The dynamical phase transition exists only for n<1. For p=2 correlation of the disorder (leading to a non-zero n) removes the known marginal stability of the spin glass phase. If the observation time is very large there occurs no finite-temperature spin glass phase. In this case there are analogies with the non-equilibrium (aging) dynamics. A generalized fluctuation-dissipation relation is derived. Received 12 July 1999 and Received in final form 8 December 1999     

An universal relation between fractal and Euclidean (topological) dimensions of random systems

It is shown that a dimension-invariant form for fractal dimension D of random systems (where d is Euclidean dimension of the embedding space) is in good agreement with results of numerical simulations performed by different authors for critical (p=p _c ) and subcritical (p<p _c ) percolation, for lattice animals, and for different aggregation processes. Received: 9 July 1998 / Revised and Accepted: 12 July 1998     

Collapse transitions of a periodic hydrophilic hydrophobic chain

We study a single self avoiding hydrophilic hydrophobic polymer chain, through Monte-Carlo lattice simulations. The affinity of monomer i for water is characterized by a (scalar) charge , and the monomer-water interaction is short-ranged. Assuming incompressibility yields an effective short ranged interaction between monomer pairs (i,j), proportional to . In this article, we take (resp. ()) for hydrophilic (resp. hydrophobic) monomers and consider a chain with (i) an equal number of hydro-philic and -phobic monomers (ii) a periodic distribution of the along the chain, with periodicity 2p. The simulations are done for various chain lengths N, in d=2 (square lattice) and d=3 (cubic lattice). There is a critical value p _c (d,N) of the periodicity, which distinguishes between different low temperature structures. For p >p _c , the ground state corresponds to a macroscopic phase separation between a dense hydrophobic core and hydrophilic loops. For p <p _c (but not too small), one gets a microscopic (finite scale) phase separation, and the ground state corresponds to a chain or network of hydrophobic droplets, coated by hydrophilic monomers. We restrict our study to two extreme cases, and to illustrate the physics of the various phase transitions. A tentative variational approach is also presented. Received: 10 March 1998 / Received in final form: 25 June 1998 / Accepted: 1st July 1998     

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

The phase transitions and critical phenomena in the three-dimensional (3D) site-diluted q-state Potts models on a simple cubic lattice are explored. We systematically study the phase transitions of the models for q=3 and q=4 on the basis of Wolff high-effective algorithm by the Monte–Carlo (MC) method. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.00–0.65. It is shown that introducing of weak disorder (p∼0.95) into the system is sufficient to change the first order phase transition into a second order one for the 3D 3-state Potts model, while for the 3D 4-state Potts model, such a phase transformation occurs when introducing strong disorder (p∼0.65). Results for 3D pure 3-state and 4-state Potts models (p=1.00) agree with conclusions of mean field theory. The static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the exponent of the correlation radius ν are calculated for the samples on the basis of finite-size scaling theory.     

Surface state on first-order ferroelectrics

In the presence of a surface the Landau-Devonshire equations of ferroelectricity must be extended to include a boundary condition. For a ferroelectric with a second-order transition in the case when the polarization p(z) increases at the surface, it is well known that a surface state occurs in a range of temperature above the bulk critical temperature t_CB . Here we explore the corresponding effect for a first-order ferroelectric. We show that a surface state can occur, but only if the surface effect is sufficiently strong. Analytic expressions are derived and illustrated for p(z), the surface value p_S =p(0) and the free energy. The transition from the paraelectric state (p=0) to the surface state is first order, and for completeness we establish the dependence of the three critical temperatures (supercooling, thermodynamic and superheating) on a boundary-condition parameter y. In a final section, we derive and illustrate expressions for p(z)in the temperature range t<t_CB .