Generic deformation channels for critical Fermi surfaces in the collisionless regime

Using a quantum Boltzmann equation framework, we analyze the nature of generic low-energy deformations of a critical surface, which exists at the non-Fermi liquid fixed point of a system consisting of fermions interacting with massless bosons. The non-Fermi liquid behaviour arises due to the itinerant quasiparticles of the Fermi surface interacting strongly with the massless bosons, which on the other hand undergo Landau damping as result of the mutual interactions. Focussing on the collisionless regime, where we neglect the collision integral, we chalk out the possible excitations spanning the entire spectrum of angular momentum ($\ell$) channels (i.e., including both small and large values of $\ell$). The excitations are of two types: particle-hole like localized excitations forming an energy band (or continuum) and delocalized collective modes with discrete energies. Although we find a collective mode analogous to the zero sound of a Fermi liquid, its dispersion shows a crossover from a $\Omega \sim |\mathbf q|^{6/5}$ behaviour to the usual $ \Omega \sim |\mathbf q|$ dependence, where $\Omega$ and $\mathbf q$ represent the frequency and momentum, respectively. We estimate the frequency scale at which this crossover takes place. We also determine the boundary for the particle-hole continuum in the $\Omega$-$\mathbf q$ plane, and observe a crossover from $\Omega \sim |\mathbf q|^{3/2}$ to $ \Omega \sim |\mathbf q|$ behaviour, determined by another crossover frequency scale.Comment: 15 pages, 4 figures; follow up paper of arXiv:2108.0948