Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

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_\rm i ) with zero mean and variance \langle h_\rm i^\rm 2 \rangle = H_\rm RF^\rm 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_\rm C = J / k and when H_\rm RF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (H_\rm 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_\rm RF increases. The ferromagnetic (FM)--paramagnetic (PM) phase boundary is clearly observed only when z \to \infty . While FM--PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (\chi ) shows a sharp cusp at T_\rm C in a small random field for finite z and rounded different peaks on increasing H_\rm RF.