Non-Fermi liquids and the Wiedemann-Franz law

A general discussion of the ratio of thermal and electrical conductivities in non-Fermi liquid metals is given. In metals with sharp Drude peaks, the relevant physics is correctly organized around the slow relaxation of almost-conserved momenta. While in Fermi liquids both currents and momenta relax slowly, due to the weakness of interactions among low-energy excitations, in strongly interacting non-Fermi liquids typically only momenta relax slowly. It follows that the conductivities of such non-Fermi liquids are obtained within a fundamentally different kinematics to Fermi liquids. Among these strongly interacting non-Fermi liquids we distinguish cases with only one almost-conserved momentum, which we term quasi-hydrodynamic metals, and with many patchwise almost-conserved momenta. For all these cases, we obtain universal expressions for the ratio of conductivities that violate the Wiedemann-Franz law. We further discuss the case in which long-lived “cold” quasiparticles, in general with unconventional scattering rates, coexist with strongly interacting hot spots, lines, or bands. For these cases, we characterize circumstances under which non-Fermi liquid transport, in particular a linear in temperature resistivity, is and is not compatible with the Wiedemann-Franz law. We suggest the likely outcome of future transport experiments on CeCoIn${}_5$, YbRh${}_2$Si${}_2$, and Sr${}_3$Ru${}_2$O${}_7$ at their critical magnetic fields.

Figure 1

Ratio of conductivities as a function of temperature in conventional metals and in certain non-Fermi liquids. Wiedemann-Franz is satisfied in regions I and III, but not in the intermediate region II. In conventional metals the crossover from II to III occurs at the Debye temperature. The non-Fermi liquids under discussion exhibit the same structure at very low temperatures.