The Viscous Flatland

Electrons in a metal usually scatter off impurities and phonons — the lattice and its defects determine how current flows. Electron hydrodynamics is the regime where electron-electron collisions dominate instead: the electrons interact with each other more than with anything else, and the current flows like a viscous fluid rather than a resistive gas. Achieving this regime requires that the electron-electron scattering length be shorter than both the impurity scattering length and the device dimensions. In ordinary metals this is difficult — the interactions are too weak relative to disorder.

Zhang et al. (arXiv:2603.11175, 2026) reach the extreme hydrodynamic regime in biased bilayer graphene by flattening the electronic bands. A perpendicular electric field opens a gap in bilayer graphene and simultaneously increases the effective electron mass. Heavier electrons interact more strongly because they move slower — the interaction time per collision is longer. In the flat band limit, electron-electron collision lengths approach 50 nanometers, comparable to the quantum wavelength itself.

Using a scanning superconducting sensor to image local current patterns, the authors map three distinct transport regimes as carrier density and electric field vary: ballistic (collisions rare, current follows geometry), hydrodynamic (collisions frequent, current flows viscously with parabolic Poiseuille profiles), and diffusive (disorder-dominated). The transitions between regimes are visible in the current maps — the spatial pattern of flow directly reveals which scattering mechanism dominates.

The structural point: the flat band does not merely slow the electrons. It changes the hierarchy of scattering mechanisms. In a dispersive band, kinetic energy dominates interactions — electrons fly past each other. In a flat band, interactions dominate kinetics — electrons are stuck together long enough to thermalize. The band structure determines not which physics is present but which physics matters. All three scattering mechanisms operate simultaneously; the flat band changes which one wins.