Exact Results for the Out of Plane Dynamical Correlation Function in a Classical Easy Plane Ferromagnet at Low Temperatures

Projection operator methodsl reduce the problem of calculating the spin density dynamical correlation function of spin systems to that of calculating the spin current relation function \( {\rm{\gamma }}_{\rm{q}}^{\rm{a}}\left( \omega \right), \) , by means of the exact result $$ {\rm{R}}_{\rm{q}}^\alpha \left( {\rm{z}} \right){\rm{ = }}\int_{\rm{0}}^\infty {{{\rm{d}}^{{\rm{izt}}}}} {\rm{ dt = }}{{{\rm{i }}} \over {{\rm{z - \omega }}_{\rm{q}}^{\alpha {\rm{,2}}}{\rm{/}}\left( {{\rm{z + \gamma }}_{\rm{q}}^{\rm{z}}\left( {\rm{z}} \right)} \right)}}{\rm{.}} $$ (1) where \( {\rm{\omega }}_{\rm{q}}^{{\rm{\alpha ,2}}} \) is the exact second moment \({\rm{ / }}\) , and denotes an equilibrium average. For classical systems below tneir lower critical dimension, the coherence length diverges as T → O. One expects to observe well defined spin waves at all wave vectors at T = 0, which implies that \({\rm{\gamma }}_{\rm{q}}^{\rm{\alpha }}\left( {\rm{\omega }} \right) \to 0.\) → O. One can expand \({\rm{\gamma }}_{\rm{q}}^{\rm{\alpha }}\left( {\rm{\omega }} \right)\) as a power series in the temperature, $$ {\rm{\gamma }}_{\rm{q}}^{\rm{\alpha }}\left( {\rm{\omega }} \right) = {\rm{KT\gamma }}_{{\rm{1,q}}}^{\rm{\alpha }}\left( {\rm{\omega }} \right){\rm{ + }}{\left( {{\rm{KT}}} \right)^{\rm{2}}}{\rm{\gamma }}_{{\rm{2,q}}}^{\rm{\alpha }}\left( {\rm{\omega }} \right){\rm{ + }} \ldots $$ (2) and, barring singularities in the higher order terms the dynamics will be determined by \({\rm{\gamma }}_{{\rm{1,q}}}^{\rm{\alpha }}\left( {\rm{\omega }} \right)\) or perhaps if this vanishes, \({\rm{\gamma }}_{{\rm{2,q}}}^{\rm{\alpha }}\left( {\rm{\omega }} \right)\) . These, however, can be calculated at T = 0, where the spin wave theory gives an exact picture of the small amplitude dynamics. The first calculation of this kind made use of an equation of motion approach for the dynamics of the operators appearing in the definition of \({\rm{\gamma }}_{\rm{q}}^{\rm{\alpha }}\left( {\rm{\omega }} \right)\) ., i.e., $$ {\rm{\gamma }}_{\rm{q}}^{\rm{\alpha }}\left( {\rm{\omega }} \right) = {\rm{ / ,}} $$ (3) where Q isαa projection operator that projects out the part of an operator that contains \({\rm{S}}_{\rm{q}}^{\rm{\alpha }}\) or \({\rm{S}}_{\rm{q}}^{\rm{\alpha }}\) , and L is the Liouville operator for the system. It was subsequently realized that a straightforward spin wave calculation led to the same results for the Heisenberg model.2 The spin wave theory is applicable, I believe, to any system below its lower critical dimension. A treatment has been given for the two dimensional Heisenberg2 model and for the in plane motion of the easy plane ferromagnet,3 $$ {\rm{H = - J}}\mathop \sum \limits_{\rm{i}} \left( {{{{\rm{\vec S}}}_{\rm{i}}} \cdot {{{\rm{\vec S}}}_{{\rm{i + 1}}}}{\rm{ + D}}{{\left( {{\rm{S}}_{\rm{i}}^{\rm{z}}} \right)}^{\rm{2}}}} \right){\rm{.}} $$ (4) The results of this theory agree with that of the isotropic model as D → O, and the theories of Villain,4 and Nelson and Fisher5 at long wavelengths. They disagree with calculations by Cieplak and Sjolander6 based upon the equation of motion method, although the two results should agree in principle. I attribute this to numerical errors in the final evaluation of their results.