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Phase diagram and thermodynamic parameters of the random field Ising model (RFIM) on spherical lattice are studied by using mean field theory. This lattice is placed in an external magnetic field (B). The random field (h i ) is assumed to be Gaussian distributed with zero mean and a variance ⟨The free energy (F ), the magnetization (M ) and the order parameter (q) are calculated. The ferromagnetic (FM) spin-glass (SG) phase transition is clearly observed. The critical temperature (T C ) is computed under a critical intensity of random field H RF = √ 2/πJ. The phase transition from FM to paramagnetic (PM) occurs at T C = J/k in the absence of magnetic field. The critical temperature decreases as H RF increases in the phase boundary of FM-to-SG. The magnetic susceptibility (χ) shows a sharp cusp at T C and the specific heat (C) has a singularity in small random field. The internal energy (U ) has a similar behaviour to that obtained from the Monte Carlo simulation.

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (h i) with zero mean and variance 〈h 2 i〉 = H RF 2 is considered. The free-energy (F), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature T C = J/k and when H RF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (H RF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field H RF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z → ∞. While FM—PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at T C in a small random field for finite z and rounded different peaks on increasing H RF.

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A one-dimensional system of six coupled random field Ising spins is studied by Glauber dynamics. The nearest-neighbour and next-nearest-neighbour interactions are taken into consideration. Two distributions of random fields (RF) -binary and Gaussian distribution -are investigated. We consider four cases of exchange couplings: ferro-ferromagnetic (F-F), ferro-antiferromagnetic (F-AF), antiferro-ferromagnetic (AF-F) and antiferro-antiferromagnetic (AF-AF). The system is fully frustrated, undergoes a zerotemperature phase transition and has multiple local energy minima in both distributions. The dynamics of the four systems are solved exactly in both distributions of RF. The effects of random fields are discussed. The number of diverging relaxation times is equal to the number of energy minima minus one. The longest relaxation times verify the Arrhenius law with energy barrier determined by the energy needed to invert the ground-state spin configuration. At low temperatures, according to the Arrhenius law, the spectrum of relaxation times shows a double-peak distribution on a logarithmic scale. In the Gaussian distribution of RF the energy-barrier distribution is continuous while it is quasi-discrete in the binary distribution.