Generic deformation channels for critical Fermi surfaces including the impact of collisions

This paper constitutes a sequel to our theoretical efforts to determine the nature of generic low-energy deformations of the Fermi surface of a quantum-critical metal, which arises at the stable non-Fermi liquid (NFL) fixed point of a quantum phase transition. The emergent critical Fermi surface, arising right at the Ising-nematic quantum critical point (QCP), is a paradigmatic example where an NFL behaviour is induced by the strong interactions of the fermionic degrees of freedom with those of the bosonic order parameter. It is an artifact of the bosonic modes becoming massless at the QCP, thus undergoing Landau-damping at the level of one-loop self-energy. We resort to the well-tested formalism of the quantum Boltzmann equations (QBEs) for identifying the excitations. While in our earlier works, we have focused on the collisionless regime by neglecting the collision integral and assuming the bosons to be in equilibrium, here we embark on a full analysis. In particular, we take into account the bosonic part of the QBEs as well, which, however, turn out to have no effect on the solutions. Decomposing the master equation into angular-momentum ($\ell$) channels, the emergent modes are of two types: Fermi-surface deformations with discrete spectra and particle-hole excitations forming a continuous band. The long-lived zero-sound mode, which corresponds to $\ell = 0$, is found to be robust against damping effects. Intriguingly, we have an infinite family of discrete modes corresponding to higher-order harmonics of the net deformation.