Fig. 9: Properties of the l = 1 longitudinal mode.
From: Hidden and mirage collective modes in two dimensional Fermi liquids
a A sketch of the trajectories of the poles of \({\chi }_{1}^{{\rm{c}}({\rm{s}})}(s)\) on the physical and unphysical Riemann surfaces. Solid (dashed) circles denote the poles on the physical (unphysical) Riemann sheet. Arrows on solid (dashed) blue lines denote the direction of poles' motion on the physical (unphysical) sheet with increasing \({F}_{1}^{{\rm{c}}({\rm{s}})}\). Blue, magenta, orange, and green circles show typical positions of the poles for the cases of an overdamped ZS mode, a hidden mode, a propagating ZS mode, and a mirage mode, respectively. Note the existence of poles (orange and green, on the \({\rm{Im}}(s)\) axis) corresponding to additional overdamped ZS modes for \({F}_{1}^{{\rm{c}}({\rm{s}})}\, > \, 0\). b A crossover in \({\chi }_{1}^{{\rm{c}}({\rm{s}})}(q,t)\) between the regions dominated by the contributions from the visible and hidden poles. The blue (magenta) points denote the numerical result for \({F}_{1}^{{\rm{c}}({\rm{s}})}={F}_{1}^{{\rm{vis}}}+0.05\) (\({F}_{1}^{{\rm{vis}}}-0.05\)), where \({F}_{1}^{{\rm{vis}}}=-0.162\), and the solid lines depict the analytical result. (The significance of \({F}_{1}^{{\rm{vis}}}\) is described in the text around Eq. (61)). It can be seen that the two traces begin in phase, then move out of phase, and finally become in-phase again. This is an indication that \({\chi }_{1}^{{\rm{c}}({\rm{s}})}(q,t)\) oscillates at different frequencies that correspond to poles for different \({F}_{1}^{{\rm{c}}({\rm{s}})}\), until oscillations from the branch points take over at long times.
