Boson theory of angular momentum: Heisenberg ferromagnetism at low temperatures

Abstract

The representation of angular momentum in terms of the boson operators originally introduced by Holstein and Primakoff is studied, and the resulting formalism is used to evaluate the low-temperature properties of the Heisenberg ferromagnet. As in the Holstein-Primakoff method, the one-to-one correspondence between spin-deviation number in the spin space and occupation number, n r, in the proper boson subspace (0 ⩽ n r ⩽ 2 S) is exploited to obtain boson representations of spin operators, and the matrix elements of these operators between improper boson states ( n r > 2 S) are treated as somewhat arbitrary adjustable parameters. Each spin operator in the boson representation may be split into a sum of proper and improper parts, and all matrix elements of the improper parts vanish within the proper subspace. Since the proper parts of the spin operators yield the correct commutation rules and matrix elements within the proper subspace, they constitute a transformation equivalent to that of Holdtein and Primakoff in which the square roots have been replaced by finite series without any special assumptions of large spin and/or low density. In general, the boson Hamiltonian may also be split into a sum of proper and improper parts, and the matrix elements of the proper part are in one-to-one correspondence with the matrix elements of the spin Hamiltonian within the proper subspace. The matrix elements of the improper part vanish within the proper subspace, and are arbitrary parameters within the improper subspace. The improper matrix elements may be chosen so as to obtain an arbitrarily large energy gap between the ground state of the spin system and the lowest improper state; in particular, the energy gap may be infinite, since the actual spin system cannot be excited into an improper state, and in this way we make contact with the hard-core model of Morita and Van Kranendonk. Finally, it is shown that in the limit of large spin the boson Hamiltonian used here reduces in form to a hermitian equivalent (H D + H D + ) 2 of Dyson's Hamiltonian, H D , thus providing a hermitian Hamiltonian which may be used to describe the Heisenberg ferromagnet with arbitrary spin when the exclusion principle for spin deviations is ignored. In applications, there are two important features of this formulation: First, since the interactions among spin waves are treated purely mechanically there are no cumber-some restrictions on the boson partition function; second, since the lowest improper boson state is separated from the ground state of the spin system by an arbitrarily large energy gap, the contribution of improper states to the ensemble average of all physical quantities may be made arbitrarily small. Without ignoring any interactions, this formulation is used to study the low-temperature properties of the Heisenberg ferromagnet. The two-particle bound states of Wortis and Hanus and the ideal scattering states of Dyson are reproduced for arbitrary spin, and moreover these results are found to be completely independent of all improper matrix elements. In evaluating the dynamical properties, we obtain off-shell terms in the boson self-energy which do not appear in previous theories, and theses are associated with the exclusion principle for spin deviations, albeit they are not exponentially small. However, it is shown that, to first order in the density of spin waves, these off-shell terms do not modify the quasiparticle spectrum found when the exclusion principle is ignored. In fact, by showing that all purely off-shell terms mutually cancel in the expression for the transverse susceptibility, we prove explicitly that the exclusion principle plays no role in the low-temperature dynamics of spin waves, at least to first order in the density. This result is a generalization of Dyson's proof that the kinematical interaction may be ignored in evaluating the low-temperature thermodynamics. Finally, we confirm that the purely off-shell terms do not contribute to the free energy, F, and thereby obtain Dyson's original result for F, including exponentially small terms.