Vortical Currents and Reciprocal Relations for Transport Coefficients in the Electron Hydrodynamic Regime

We investigate the hydrodynamic regime in metals with momentum-conserving electron-electron

scattering. The conservation of momentum results in well-defined dynamics whose effects we investigate

via the relevant continuity equations. We find anomalous contributions to the charge and heat transport

currents arising from gradients of the velocity field in a semiclassical treatment with a Berry curvature.

These contributions are nonvanishing for systems lacking inversion symmetry, and the corresponding

transport coefficients do not obey the standard Onsager reciprocity relations. Instead, we show that the

response coefficients relating the currents to the stress tensor obey independent reciprocity relations with

the stress tensor and thus exhibit cross-tensor effects of charge and heat transport with the momentum

transport. The Berry curvature contribution to the stress magnetization tensor is also obtained.