Perturbation theory for two-dimensional hydrodynamic plasmons

Perturbation theory is an indispensable tool in quantum mechanics and electrodynamics that handles weak effects on particle motion or fields. However, its extension to plasmons involving complex motion of both particles and fields remained challenging. We show that this challenge can be mastered if electron motion obeys the laws of hydrodynamics, as recently confirmed in experiments with ultraclean heterostructures. We present a unified approach to evaluate corrections to plasmon spectra induced by carrier drift, magnetic field, scattering, viscosity, and Berry curvature. The developed theory enables us to resolve the long-standing stability problem for direct current in confined two-dimensional electron systems against self-excitation of plasmons. We show that arbitrarily weak current in the absence of dissipation is unstable provided the structure lacks mirror symmetry. On the contrary, we find that in extended periodic systems---plasmonic crystals---weak carrier drift is always stable. Instead, this drift induces anomalous Doppler shift, which can be both below and higher than its value in uniform systems.