Kinetic Theory of Electronic Transport in Random Magnetic Fields

We present the theory of quasiparticle transport in perturbatively small inhomogeneous magnetic fields across the ballistic-to-hydrodynamic crossover. In the hydrodynamic limit, the resistivity $\rho$ generically grows proportionally to the rate of momentum-conserving electron-electron collisions at large enough temperatures $T$. In particular, the resulting flow of electrons provides a simple scenario where viscous effects suppress conductance below the ballistic value. This new mechanism for $\rho \propto T^2$ resistivity in a Fermi liquid may describe low $T$ transport in single-band ${\text{SrTiO}}_3$.

Figure 1

(a) Resistivity in the two Fermi surface model with magnetic disorder; (b) in the model Eq. (20) with magnetic disorder; (c) in the two Fermi surface model with potential disorder; (d) in the model Eq. (20) with potential disorder. Data in (c) and (d) are taken from Ref. [21]. Resistivities are normalized by ${\rho }_{\mathrm{res}}$, the ballistic residual resistivity at ${\ell }_{ee}=\infty$. Equation (18) leads to the blue ($v_{F,1}/v_{\mathrm{F},2}=1$) curve in panels (a) and (b). We observe how whether electron-electron interactions enhance or suppress transport beyond the ballistic limit is sensitive to the particular hydrodynamic limit and the nature of disorder.