Fig. 7: A 3D depiction of the pole evolution on the Riemann surface for l = 0.
From: Hidden and mirage collective modes in two dimensional Fermi liquids
The figure is obtained by mapping the complex s point of the two Riemann sheets to the 3D set of points \(\{{\rm{Re}}s,{\rm{Im}}s,\pm {\rm{Re}}\sqrt{1-{s}^{2}}\}\) where +(−) maps the physical (unphysical) sheet to the top (bottom) sheet of the figure. On this representation of the surface, the solid and blue red lines denote the pole evolution with increasing \({F}_{0}^{{\rm{c}}({\rm{s}})}\). The evolution of the poles begins at the origin of the physical and unphysical sheets at \({F}_{0}^{{\rm{c}}({\rm{s}})}=-1\). The poles initially move along Im(s) axis down(up) the physical(unphysical) sheets. The pole on the unphysical sheet reaches infinity and crosses to the physical sheet at \({F}_{0}^{{\rm{c}}({\rm{s}})}=-1/2\), and the poles merge and bifurcate at \({F}_{0}^{{\rm{c}}({\rm{s}})}=-(1-{\gamma }^{2})/2\). The regions with yellow shading denote areas where a pole in \({\chi }_{0}^{{\rm{c}}({\rm{s}})}(s)\) either on physical, or on unphysical Riemann sheet, gives rise to a peak in \({\chi }_{0}^{{\rm{c}}({\rm{s}})}(s)\) on the physical real s axis. The areas shaded by peach color are regions where a pole cannot be analytically extended to the physical real axis due to the branch cuts, and \({\chi }_{0}^{{\rm{c}}({\rm{s}})}(s)\) on the physical real frequency axis has no sharp peaks. We set γ = 0.2 for definiteness.
