关键词: random field, replica trick, coordination number, order parameter, susceptibility and phase diagram

Abstract: 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 $H_{\rm RF}$.

Key words: random field, replica trick, coordination number, order parameter, susceptibility and phase diagram