Contribution to the solution of Heisenberg's model for phase transitions: The A2→B2 disorder-order transition N binary alloys of AB stoichiometry

Abstract

A fictitious spin operator is introduced in the state space of a lattice site i of a binary alloy. The spin operators of an isolated pair of nearest-neighbour sites i and j are coupled by the Heisenberg exchange Hamiltonian. Introduction of the Gibbs grand canonical ensemble enables us to show that the interaction Hamiltonian between the pair ij and the rest of the crystal may be written simply as Ĥ int = - μ N̂ (where N̂ is the particle number operator corresponding to the long-range- order parameter and μ is its intensive conjugate variable). The state vector and the density matrix of the pair ij are then obtained by diagonalizing the operator corresponding to the grand canonical potential. Among others, the grand partition function is obtained as a function of the intensive variables of the problem. Neither the symmetry of the state vector nor the long-range-order parameter is a constant of the motion; they fluctuate with time in the vicinity of the transition.