Scale Invariance and Self-averaging in disordered systems

In a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.Comment: 7 pages, 4 ps figure

Stability of critical behaviour of weakly disordered systems with respect to the replica symmetry breaking

A field-theoretic description of the critical behaviour of the weakly disordered systems is given. Directly, for three- and two-dimensional systems a renormalization analysis of the effective Hamiltonian of model with replica symmetry breaking (RSB) potentials is carried out in the two-loop approximation. For case with 1-step RSB the fixed points (FP's) corresponding to stability of the various types of critical behaviour are identified with the use of the Pade-Borel summation technique. Analysis of FP's has shown a stability of the critical behaviour of the weakly disordered systems with respect to RSB effects and realization of former scenario of disorder influence on critical behaviour.Comment: 10 pages, RevTeX. Version 3 adds the $\beta$ functions for arbitrary dimension of syste

Critical behavior of disordered systems with replica symmetry breaking

A field-theoretic description of the critical behavior of weakly disordered systems with a $p$-component order parameter is given. For systems of an arbitrary dimension in the range from three to four, a renormalization group analysis of the effective replica Hamiltonian of the model with an interaction potential without replica symmetry is given in the two-loop approximation. For the case of the one-step replica symmetry breaking, fixed points of the renormalization group equations are found using the Pade-Borel summing technique. For every value $p$, the threshold dimensions of the system that separate the regions of different types of the critical behavior are found by analyzing those fixed points. Specific features of the critical behavior determined by the replica symmetry breaking are described. The results are compared with those obtained by the $\epsilon$-expansion and the scope of the method applicability is determined.Comment: 18 pages, 2 figure

The Wandering Exponent of a One-Dimensional Directed Polymer in a Random Potential with Finite Correlation Radius

We consider a one-dimensional directed polymer in a random potential which is characterized by the Gaussian statistics with the finite size local correlations. It is shown that the well-known Kardar's solution obtained originally for a directed polymer with delta-correlated random potential can be applied for the description of the present system only in the high-temperature limit. For the low temperature limit we have obtained the new solution which is described by the one-step replica symmetry breaking. For the mean square deviation of the directed polymer of the linear size L it provides the usual scaling $L^{2z}$ with the wandering exponent z = 2/3 and the temperature-independent prefactor.Comment: 14 pages, Late

Numerical Results For The 2D Random Bond 3-state Potts Model

We present results of a numerical simulation of the 3-state Potts model with random bond, in two dimension. In particular, we measure the critical exponent associated to the magnetization and the specific heat. We also compare these exponents with recent analytical computations.Comment: 9 pages, latex, 3 Postscript figure

Critical behavior of the pure and random-bond two-dimensional triangular Ising ferromagnet

We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios - logarithmic corrections versus weak universality - strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality.Comment: 10 pages, 8 figures, slightly revised version as accepted for publication in Phys. Rev.

Coupled Ising models with disorder

In this paper we study the phase diagram of two Ising planes coupled by a standard spin-spin interaction with bond randomness in each plane. The whole phase diagram is analyzed by help of Monte Carlo simulations and field theory arguments.Comment: 9 pages and 3 figure

Non-perturbative phenomena in the three-dimensional random field Ising model

The systematic approach for the calculations of the non-perturbative contributions to the free energy in the ferromagnetic phase of the random field Ising model is developed. It is demonstrated that such contributions appear due to localized in space instanton-like excitations. It is shown that away from the critical region such instanton solutions are described by the set of the mean-field saddle-point equations for the replica vector order parameter, and these equations can be formally reduced to the only saddle-point equation of the pure system in dimensions (D-2). In the marginal case, D=3, the corresponding non-analytic contribution is computed explicitly. Nature of the phase transition in the three-dimensional random field Ising model is discussed.Comment: 12 page

On the nature of the phase transition in the three-dimensional random field Ising model

A brief survey of the theoretical, numerical and experimental studies of the random field Ising model during last three decades is given. Nature of the phase transition in the three-dimensional RFIM with Gaussian random fields is discussed. Using simple scaling arguments it is shown that if the strength of the random fields is not too small (bigger than a certain threshold value) the finite temperature phase transition in this system is equivalent to the low-temperature order-disorder transition which takes place at variations of the strength of the random fields. Detailed study of the zero-temperature phase transition in terms of simple probabilistic arguments and modified mean-field approach (which take into account nearest-neighbors spin-spin correlations) is given. It is shown that if all thermally activated processes are suppressed the ferromagnetic order parameter m(h) as the function of the strength $h$ of the random fields becomes history dependent. In particular, the behavior of the magnetization curves m(h) for increasing and for decreasing $h$ reveals the hysteresis loop.Comment: 22 pages, 12 figure