Self-Consistent Selection of a Ferromagnetic Representation for the Heisenberg-Exchange Model

The Heisenberg magnetic-exchange Hamiltonian is written in second-quantized form and a $\frac{1}{V}$ factor is extracted, where $V$ is the volume of the system. Using Umezawa's self-consistent method, a unitarily inequivalent representation is selected in which the Hamiltonian obviously describes a ferromagnetic system; a result not at all obvious since the original Hamiltonian is completely symmetric and there is no reason $a$ priori for expecting it to describe an asymmetric ferromagnetic configuration. All higher-order terms are accounted for, and the representation is picked out without using the adiabatic theorem, which is typically used in the self-consistent method. Inequivalence of various representations is discussed and validity is added for using an exchange integral depending only on relative distance between lattice sites and, in particular, on nearest neighbors.