Ballistic-to-hydrodynamic transition and collective modes for two-dimensional electron systems in magnetic field

The recent demonstrations of viscous hydrodynamic electron flow in two-dimensional electron systems poses serious questions to the validity of existing transport theories, including the ballistic model, the collision-induced and collisionless hydrodynamics. While the theories of transport at hydrodynamic-to-ballistic crossover for free 2d electrons are well established, the same is not true for electrons in magnetic fields. In this work, we develop an analytically solvable model describing the transition from ballistic to hydrodynamic transport with changing the strength of electron-electron collisions in magnetic fields. Within this model, we find an expression for the high-frequency non-local conductivity tensor of 2d electrons. It is valid at arbitrary relation between frequency of external field $\omega$, the cyclotron frequency $\omega_c$, and the frequency of e-e collisions $\tau^{-1}_{ee}$. We use the obtained expression to study the transformation of 2d magnetoplasmon modes at hydrodynamic-to-ballistic crossover. In the true hydrodynamic regime, $\omega\tau_{ee} \ll 1$, the 2DES supports a single magnetoplasmon mode that is not split at cyclotron harmonics. In the ballistic regime, $\omega\tau_{ee} \gg 1$, the plasmon dispersion develops splittings at cyclotron harmonics, forming the Bernstein modes. A formal long-wavelength expansion of kinetic equations ("collisionless hydrodynamics") predicts the first splitting of plasmon dispersion at $\omega\approx 2\omega_c$. Still, such expansion fails to predict the zero and negative group velocity sections of true magnetoplasmon dispersion, for which the full kinetic model is required.