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Non-Fermi liquids and the Wiedemann-Franz law

Raghu Mahajan, Maissam Barkeshli, Sean A. Hartnoll

TL;DR

The paper develops a memory-matrix framework to analyze heat and charge transport in non-Fermi liquids, focusing on how almost-conserved momentum governs conductivities in various kinematic regimes. It identifies three key NFL classes: quasi-hydrodynamic metals with a single almost-conserved momentum, patchwise NFLs with many conserved momenta, and NFLs with coexisting long-lived quasiparticles, deriving universal or near-universal relations for the Lorenz ratio and clarifying when the Wiedemann-Franz law holds or is violated. It shows that linear-in-$T$ resistivity can coexist with WF under elastic scattering from generalized phonon-like modes, but inelastic hot-mode scattering generically violates WF; it also provides experimental benchmarks and discusses how measurements of $\kappa$ relative to $\overline{\kappa}$ can diagnose the underlying transport kinematics. The work offers a diagnostic framework for quantum critical transport applicable to heavy fermions and ruthenates, bridging weak-coupling intuition and strongly coupled transport paradigms, with implications for interpreting CeCoIn$_5$, YbRh$_2$Si$_2$, and Sr$_3$Ru$_2$O$_7$ data.

Abstract

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 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 CeCoIn5, YbRh2Si2 and Sr3Ru2O7 at their critical magnetic fields.

Non-Fermi liquids and the Wiedemann-Franz law

TL;DR

The paper develops a memory-matrix framework to analyze heat and charge transport in non-Fermi liquids, focusing on how almost-conserved momentum governs conductivities in various kinematic regimes. It identifies three key NFL classes: quasi-hydrodynamic metals with a single almost-conserved momentum, patchwise NFLs with many conserved momenta, and NFLs with coexisting long-lived quasiparticles, deriving universal or near-universal relations for the Lorenz ratio and clarifying when the Wiedemann-Franz law holds or is violated. It shows that linear-in- resistivity can coexist with WF under elastic scattering from generalized phonon-like modes, but inelastic hot-mode scattering generically violates WF; it also provides experimental benchmarks and discusses how measurements of relative to can diagnose the underlying transport kinematics. The work offers a diagnostic framework for quantum critical transport applicable to heavy fermions and ruthenates, bridging weak-coupling intuition and strongly coupled transport paradigms, with implications for interpreting CeCoIn, YbRhSi, and SrRuO data.

Abstract

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 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 CeCoIn5, YbRh2Si2 and Sr3Ru2O7 at their critical magnetic fields.

Paper Structure

This paper contains 13 sections, 32 equations, 1 figure, 1 table.

Table of Contents

  1. The Wiedemann-Franz law
  2. The memory matrix
  3. Fermi liquids
  4. Non-Fermi liquids without quasiparticles
  5. Quasi-hydrodynamic non-Fermi liquid metals
  6. Non-Fermi liquids with patchwise conserved momenta
  7. Non-Fermi liquids with (some) long-lived quasiparticles
  8. Linear in temperature resistivity and Wiedemann-Franz
  9. Revisiting the experiments
  10. Discussion: The two faces of quantum criticality
  11. Strong momentum relaxation
  12. Patchwise conserved momenta: Ising-nematic example
  13. Specific heat of generalized phonons

Figures (1)

  • 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.