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The main subject of this talk is thermal transport in the disordered Fermi and electron liquids at low temperatures. I plan to start with a brief introduction to the physics of quantum corrections to conductivity in disordered systems and the phenomenology of the metal-insulator transition in two-dimensional electron systems. Then, I will contrast approaches to the calculation of electric and thermal transport. A principle difficulty for the description of thermal transport is that temperature is an internal parameter, and a temperature gradient does not correspond to an external ""mechanical"" force like a the one originating from an electric potential. This problem becomes particularly severe when the description of low temperature phenomena requires a renormalization group (RG) analysis, which is hard to implement in a kinetic equation approach. We therefore use Luttinger's gravitational potentials as sources for finding the heat density and its correlation function. For a comprehensive study, we extend the RG analysis developed for electric transport by including the gravitational potentials into the RG scheme. The analysis reveals that for the disordered Fermi liquid the Wiedemann-Franz law remains valid even in the presence of quantum corrections caused by the interplay of diffusion modes and the electron-electron interaction. In the present scheme this fundamental relation is closely connected with a fixed point in the multi-parametric RG flow of the gravitational potentials. For the disordered electron liquid we additionally analyze inelastic processes induced by the Coulomb interaction at sub-temperature energies. While the general form of the correlation function has to be compatible with energy conservation, these inelastic processes are at the origin of logarithmic corrections violating the Wiedemann-Franz law. The interplay of various terms in the heat density-heat density correlation function therefore differs from that for densities of other conserved quantities, such as the total number of particles or spin.

References:
G. Schwiete and A. M. Finkel'stein, Renormalization group analysis of thermal transport in the disordered Fermi liquid, Phys. Rev. B 90, 155441 (2014)
G. Schwiete and A. M. Finkel'stein, Thermal transport and the Wiedemann Franz law in the disordered Fermi liquid, Phys. Rev. B 90, 060201(R) (2014)
G. Schwiete and A. M. Finkel'stein, Keldysh approach to the renormalization group analysis of the disordered electron liquid, Phys. Rev. B 89, 075437 (2014)