Spectral weight in Chern-Simons theory with symmetry breaking

We calculate the low-energy spectral weight of a holographic superfluid coupled to a Chern-Simons term in IR radial scaling geometries characterized by a parameter $\eta$. This work was motivated by previous results where an unexpected low-energy spectral weight and a region of instability were seen, both at finite momentum, for the holographic superfluid. We characterize the effect of varying the Chern-Simons coupling $\alpha$ and condensate charge parameter $\zeta$ on these regions supporting low-energy spectral weight or a finite momentum instability. We show that $\eta$, $\alpha$ and $\zeta$ each plays a unique role in shaping these regions. We find a surface $\alpha_{\text{crit}}(\eta, \zeta)$ above which the theory is unstable. In the longitudinal channel we extend our analysis to general dimension $d$. We briefly analyze the Einstein-Maxwell-dilaton theory and find that Fermi shells exist for $d>4$.