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 T c ∗ ( H ) , on the other hand in the continuous transition the magnetization is null at T = T c ( H ) . At null temperature ( T = 0 ) we have found the coexistence field H c = 3.23  J that is independent of spin value. The transition temperature T c ( H ) for the classical case ( S = ∞ ), in the T − H plane, is larger than the quantum case ( S = 1 / 2 ).