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 wave-number-dependent conductivity $\sigma \left(k\right)$, which incorporates rates of realistic microscopic scattering processes and is built as a continued fraction from decay rates for different excitations. This approach is used to clarify the relationship between nonlocal response and vortices across ballistic and hydrodynamic phases. Vorticity exhibits similar values in both phases but shows markedly 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 diagnostic of the microscopic origin of vorticity in electron fluids.