Random Field Ising Model

Two varieties of disorder are encountered in spin models, randomness in the strength of the bonds and randomness in the strength of an externally applied magnetic field. Research on Spin Glass models, which suffer from the first type of disorder, has been very sucessful. Random Field models, in which the disorder couples directly to the order parameter have proven less amenable to progress. Early work on the pertubative approach led to the prediction of dimensional reduction and culminated in the beautiful explanation of this phenomenon in terms of the supersymmetry mechanism of Parisi and Sourlas. It became clear however that this perturbative approach did not capture the complex behavior of real materials. We are still in need of new non-perturbative understanding of the physics of random field systems.

In the Complex Systems group we are working on various aspects of this problem including both simulation and analytical approaches. We have concentrated our attention on the simplest model, that of the Random Field Ising Model (RFIM).


Since Monte Carlo simulations of the RFIM are hampered by extremely long relaxation times (and there is no multi-spin cluster code to speed things up) we have been motivatated to think of other approaches to simulating the system. We have pioneered a direct approach to the mean field theory. The mean field theory (MFT) for the RFIM is not as easily understood as for the pure Ising Model because the mean field equations are complicated and in fact, below a certain temperature, have many solutions. In this case we define MFT by the intuitive method of weighting each solution with its Boltzmann factor (actualy e-Free Energy). This comes into practical use only through our ability to solve the mean field equations numerically. We have used a very efficient code on APE in order to deal with reasonably large lattice sizes. Although the correlation length is finite for each solution of the mean field equations Guagnelli, Marinari and Parisi we have shown that critical behavior can arise from the sum over solutions and have measured the resulting exponents.

Besides looking at the exponents, this method provides intuition concerning the domain structure of the theory.

For example this shows the domain-like structure that exists at temperatures above, but close to critical. For more details look at our paper .


The theory of broken replica symmetry has been very sucessful in understanding the physics of spin glasses and it is natural to investigate its consequences for random field systems. Within the context of the replica approach to the RFIM, the scenario is as follows. The replica symmetric solution becomes unstable as one reduces the temperature below TRSB, at which point the correlation length remains finite Mezard and Monasson. The critical temperature TC at which ferromagnetic order arises is at a lower temperature TC < TRSB. Finally one expects replica symmetry to be restored at an even lower temperature. Our understanding however, is limited because the form of replica symmetry breaking is unknown. An approach based on a variational method has had some sucess a the level of the effective theory of the interfaces Mezard and Parisi but this technique has not so far yielded results for the bulk problem Mezard and Young. An alternative approach is to consider the effect of instantons breaking the replica symmetry Dotsenko and Parisi and this may be rephrased in the language of supersymmetry.

David Lancaster (djl@liocorno.roma1.infn.it) 16/1/95