Generic deformation channels for critical Fermi surfaces in the collisionless regime

Using a quantum Boltzmann equation framework, we analyse the nature of generic low-energy deformations of a critical Fermi 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 a 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 ($\mathrm{\ell }$) channels (i.e., including both small and large values of $\mathrm{\ell }$). 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 $\mathrm{\Omega }\sim \mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^{6\mathrm{/}5}$ behaviour to the usual $\mathrm{\Omega }\sim \mathrm{\mid }\mathbf{q}\mathrm{\mid }$ dependence, where $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$–$\mathbf{q}$ plane, and observe a crossover from $\mathrm{\Omega }\sim \mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^{3\mathrm{/}2}$ to $\mathrm{\Omega }\sim \mathrm{\mid }\mathbf{q}\mathrm{\mid }$ behaviour, determined by another crossover frequency scale.