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Dynamical Response of Single Bi-layer Spin Model : A Theoretical Analysis

The spin dynamics of a single bi-layer ferromagnetic model, as proposed for a manganite system $La_{1.2}Sr_{1.8}Mn_{2}O_{7}$ as an approximate minimal model, is studied both below and above the bulk magnetic transition temperature using the semi-classical Monte Carlo-molecular dynamics technique. The quantities studied are:- (i) the static spin configurations, (ii) the spin auto-correlation function (SAF) $C(t)$ and (iii) the dynamical structure function (DSF) $S(\bf q, ω)$. The major aim is to probe the nature of collective modes and in particular to ascertain whether the propagating modes can exist above the ordering temperature. We find that the typical spin configurations contain a high degree of mis-aligned spins, particularly in the high temperature phase. Nevertheless, in the low temperature phase a long range ferromagnetic ordering can be seen. The shape of the curves corresponding to $S(\bf q, ω)$ vs. $ω$ in the constant $\bf q$-scan are found in general to be quite complex viz. containing peaks, cusps and broad plateau. Moreover, in a large regime of $(\bf q, ω)$ space, $S(\bf q ,ω)$ is found to be negative, signalling a total breakdown of the semi-classical approximation in the presence of the enormous thermodynamic and quantum fluctuations. Considering the physically allowed regime only, we can theoretically extract an effective $ω$ vs. $q$ dispersion curve for the collective excitations, which exhibits a slope opposite to that expected for a full three-dimensional system and disagrees with the results of inelastic neutron scattering experiments in the nature of the dispersion curve along the $<100>$ direction in the ordered phase. The theoretical results also exhibit the existence of propagating modes even in the high temperature (disordered) phase from our calculations based on a minimal single bi-layer ferromagnetic model.