Critical correlation susceptibility relation in random-field systems

Noise-to-noise ratios in correlation length calculations near criticality

For finite quenched random systems, on regular lattices, it is possible to define two types of variances (noises). It is demonstrated that their ratio is useful in calculating the correlation length of an infinite and rather general random system, as a function of temperature. The numerical method of obtaining those variables is not relevant. It can be real-space numerical renormalization, simulation, or any other method. It does not matter. The correlation length obtained by this technique may then be used to obtain directly the critical correlation exponent, ν, rather than indirectly, using scaling relations, as is often done. The method is demonstrated by applying it to the random field Ising model.

Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.

Ground states and thermal states of the random field Ising model

The random field Ising model is studied numerically at both zero and positive temperature. Ground states are mapped out in a region of random field and external field strength. Thermal states and thermodynamic properties are obtained for all temperatures using the Wang-Landau algorithm. The specific heat and susceptibility typically display sharp peaks in the critical region for large systems and strong disorder. These sharp peaks result from large domains flipping. For a given realization of disorder, ground states and thermal states near the critical line are found to be strongly correlated--a concrete manifestation of the zero temperature fixed point scenario.

Full reduction of large finite random Ising systems by real space renormalization group

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain Gamma(r)=<sigma(0)sigma(r)>-<sigma(0)><sigma(r)>, the "connected" correlation function, and S(r)=<sigma(0)sigma(r)>, the "disconnected" correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function Gamma(r) along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such nu or eta). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.

Critical exponents of the random-field O(N) model

The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.