Study of the first-order phase transition in the classical and quantum random field Heisenberg model on a simple cubic lattice

The phase diagram of the Heisenberg ferromagnetic model in the presence of a magnetic random field (we have used bimodal distribution) of spin S=1/2 (quantum case) and S=∞ (classical case) on a simple cubic lattice is studied within the framework of the effective-field theory in finite cluster (we have chosen N=2 spins). Integrating out the part of order parameter (equation of state), we obtained an effective Landau expansion for the free energy written in terms of the order parameter Ψ(m). Using the Maxwell construction we have obtained the phase diagram in the T−H plane for all intervals of the field. The first-order transition temperature is calculated by the discontinuity of the magnetization at Tc∗(H), on the other hand in the continuous transition the magnetization is null at T=Tc(H). At null temperature (T=0) we have found the coexistence field Hc=3.23 J that is independent of spin value. The transition temperature Tc(H) for the classical case (S=∞), in the T−H plane, is larger than the quantum case (S=1/2).