Effect of elastic deformations on the critical behavior of disordered systems with long-range interactions

A field-theoretic approach is applied to describe behavior of three-dimensional, weakly disordered, elastically isotropic, compressible systems with long-range interactions at various values of a long-range interaction parameter. Renormalization-group equations are analyzed in the two-loop approximation by using the Padé-Borel summation technique. The fixed points corresponding to critical and tricritical behavior of the systems are determined. Elastic deformations are shown to changes in critical and tricritical behavior of disordered compressible systems with long-range interactions. The critical exponents characterizing a system in the critical and tricritical regions are determined.     

Influence of long-range effects on the critical behavior of three-dimensional systems

The Padé-Borel resummation technique is used to describe field-theoretically, in the two-loop approximation, the behavior of Ising systems with long-range effects directly in a three-dimensional space. The renormalization-group equations are analyzed and the fixed points governing the critical behavior of the system are determined. It is shown that the long-range effects can bring about a change in both the regime of critical behavior and the kind of phase transition.     

Effects of long-range interaction on critical and multicritical behavior of compressible disordered systems

A field-theoretic approach is applied to describe behavior of weakly disordered, isotropic elastic compressible systems with long-range interactions directly in the three-dimensional space for various values of the long-range interaction parameter a. A renormalization-group procedure is applied separately for a > 2 and a ≤ 2 directly in the three-dimensional space. Renormalization-group equations are analyzed in the two-loop approximation, and critical and tricritical points are determined. It is shown that long-range effects are not important when a ≤ 2, whereas they play a key role in the opposite case of a > 2. Critical exponents characterizing the system are obtained for various values of the long-range interaction parameter. Behavior of homogeneous and disordered systems characterized by two fluctuating order parameters is also described.     

Boundary between long-range and short-range critical behavior in systems with algebraic interactions

We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r(-15/4). This answers a long-standing controversy between mutually conflicting renormalization-group analyses.     

Effect of long-range interactions on the multicritical behavior of homogeneous systems

A field-theoretic approach is applied to describe behavior of homogeneous three-dimensional systems with long-range interactions defined by two order parameters at bicritical and tetracritical points. Renormalization-group equations are analyzed in the two-loop approximation by using the Padé-Borel summation technique. The fixed points corresponding to various types of multicritical behavior are determined. It is shown that effects due to long-range interactions can be responsible for a change from bicritical to tetracritical behavior.     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

Anisotropic,long-range elastic interaction between adatoms

The interaction between two adatoms mediated by the elastic distortion of the substrate is calculated using the Green's functions derived by Maradudin et al. for semi-infinite hexagonal and cubic crystals. For a substrate with hexagonal symmetry with the surface parallel to the basal plane, the interaction energy is isotropic over the substrate surface, and varies as R⁻³ with the separation R of the adatoms, as in the case of an elastically isotropic substrate. The interaction between adatoms on the (100) face of cubic crystal also varies as R⁻³, but is anisotropic. In many cases, the interaction between identical adatoms is attractive along the cube axes and repulsive along directions at 45° to the axes.     

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in . Received 9 June 2000 and Received in final form 2 August 2000     

Critical behavior in random-spin systems with long-range interactions

Effects of the potential range of the interaction to critical behaviors of quenched random-spin systems are investigated in the limit M → 0 of the MN-component models by means of the renormalization-group theories. As static critical phenomena the stability of the fixed points is investigated and the critical exponents (η, γ, , crossover index) and the equation of state are derived. These phenomena are different from those in pure systems, for the positive specific heat exponent of the pure Heisenberg system.     

Critical behavior in anisotropic cubic systems with long-range interactions

Effects of the potential range of interaction to critical behaviors of anisotropic cubic systems are investigated by means of the Callan-Symanzik equations. As the static critical behavior the stability of fixed points, the critical exponents η^C, γ^C, φ^C_s and φ^C_c, and the equation of state are also investigated. As the dynamic critical behavior the dynamic critical exponent z_φ is derived based on the time-dependent Ginzburg-Landau stochastic model. The two- and three-dimensional critical behaviors are discussed.     

Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.