Kinematical problem in Spin-wave theory

A spin-1/2, nearest neighbor Heisenberg Hamiltonian acting on a periodic,d-dimensional lattice is considered. Multi-spin-wave solutions to the Schrödinger equation for a Heisenberg ferromagnet involve an unlimited superposition of spin-reversal operators at sites. This violates the physical restriction that no more than one excitation reside on any one site. This exclusion rule affects spin-wave interaction—the determination of these effects is called the kinematical problem. A general nonperturbative treatment that includes kinematical effects in spin-wave theory is developed along the following lines. Using the property of the Heisenberg Hamiltonian that it does not couple states obeying the single occupation condition at all sites with states that violate the single-occupancy condition at some sites, the unphysical multiply occupied states can be eliminated by a nonunitary transformation of the eigenvalue equation. An overcomplete Hamiltonian matrix is obtained that contains all the physical eigenvalues as a subset of its spectrum. Overcompleteness is shown to be a large part of the kinematical problem and several schemes to handle it are discussed. The remainder of the kinematical problem lies in the nonorthogonality of spin waves. It is shown that a new type of distribution, one that is neither Bose nor Fermi, correctly describes free spin-wave statistics at all temperatures. This formal but nonetheless complete solution to the overcompleteness aspect of the kinematical problem is then carried over,in toto, to the boson formulation of the spin Hamiltonian. Application to the calculation of the partition function and to thermal Green's functions is noted.