Spin-wave renormalization in the Heisenberg ferromagnet

Abstract

The cumulative effects of higher-order correlations and spin-wave interactions are represented in the usual versions of a practical (first-order) Green's-function theory of the Heisenberg ferromagnet by the renormalization of the spin-wave energies. The problem of deriving the appropriate renormalization factor $R$ in a given temperature regime based on a comparison with some exact result is considered. To this end, Dyson's rigorous asymptotic series for the spontaneous magnetization $\sigma$ and the free energy at very low temperatures are used as a "boundary condition" on the theory to derive necessary conditions on $R$. Spectral relations are then used to derive an expression for $R$ of the form $h^{\frac{(1+\xi )}{(2+\xi )}}$, where $h$ is the average energy $〈\mathcal{H}〉$ of the system measured in units of the ground-state energy; 1, $\xi$ are certain relative contributions to the $T^4$ term in $\sigma$ and represent the leading effects of the dynamical interaction of spin waves, respectively, in the Born approximation and in higher orders. The phenomenological "second random-phase approximation" is a special case of this form, corresponding to the retention of only the Born approximation in the above. The explicit occurrence of $R$ (due to a summation over renormalized spin waves) in the spectral relation for the Hamiltonian is exploited to eliminate $h$ and find a result for $R$ in terms of the customary parameters of a first-order theory, that is automatically "moment conserving." From this result, other expressions for $R$ that are equivalent to it at low temperatures, including that of Callen's decoupling scheme, are derived. The differences and difficulties that arise in the special case $S=\frac{1}{2}$ are brought out clearly and discussed. The appropriate modulation that the $R$ factor of the randomphase approximation must undergo in this case in order to lead to the correct low-temperature series for $\sigma$ is deduced. It is also proved that it is impossible for a linearized Green's-function theory for the $S=\frac{1}{2}$ ferromagnet to yield correct results at low $T$ for both $\sigma$ and the specific heat if the theory is of the pure pole type (the Green's function is given by a magnon pole term alone) with a wave-vector-independent renormalization of the spin wave spectrum: it is necessary to have at least a $k$-dependent $R$, or a dispersive part in addition to the pole term. Writing the spectral relation for the Hamiltonian in terms of the spin-spin correlation functions, is is shown that theories of the above kind (pole type, with $k$-independent $R$) also suffer from a serious defect near $T_C$, for all $S$. The central role of the longitudinal correlation is emphasized and the conditions necessary for its proper determination to ensure a consistent linearized theory are discussed. The detailed derivation of a theory with the requisite characteristics will be presented in another paper.