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

TL;DR

This study investigates the phase diagram and first-order phase transition behavior of classical and quantum Heisenberg models with random magnetic fields on a cubic lattice, revealing a coexistence field and differences between classical and quantum cases.

Contribution

It introduces an effective-field theory approach with finite clusters to analyze the phase transitions in the random field Heisenberg model, including both classical and quantum spins.

Findings

Coexistence field at H_c=3.23J independent of spin value.

Classical case exhibits higher transition temperatures than quantum case.

First-order transition characterized by magnetization discontinuity.

Abstract

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=\infty $ (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 $\Psi (m)$. Using Maxwell construction we have obtained the phase diagram in the $T-H$ plane for all interval of field. The first-order transition temperature is calculated by the discontinuity of the magnetization at $T_{c}^{\ast}(H)$, 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 \textbf{coexistence} field $H_{c}=3.23J$ that is independent of spin value. The transition temperature $T_{c}(H)$ for the classical case ($S=\infty $), in the $T-H$ plane, is larger than the quantum case (S=1/2).