Nonlocal conductivity, continued fractions and current vortices in electron fluids

Vortices in electron fluids are a key indicator of electron hydrodynamics. However, a comprehensive framework linking macroscopic vorticity measurements with microscopic interactions and scattering mechanisms has been lacking. We employ wavenumber-dependent conductivity $\sigma(k)$ incorporating realistic microscopic scattering processes, aiming to clarify the relationship between nonlocal response and vortices across ballistic and hydrodynamic phases. Vorticity is found to take similar values in both phases but feature very different sensitivity to momentum-relaxing scattering, with ballistic vortical flows being orders-of-magnitude more resilient than the hydrodynamic ones. This behavior can serve as a simple diagnostic of the microscopic origin of vorticity in electron fluids.