Quantum kinetic approach for studying thermal transport in the presence of electron-electron interactions and disorder

A user friendly scheme based on the quantum kinetic equation is developed for studying thermal transport phenomena in the presence of interactions and disorder. We demonstrate that this scheme is suitable for both a systematic perturbative calculation as well as a general analysis. We believe that we present an adequate alternative to the Kubo formula, which for the thermal transport is rather cumbersome.

Figure 1

(a) Illustration of Eq. (8a) for the full Green’s function $\widehat{G}$. (b) The Dyson equation for ${\widehat{G}}_{int}$ [see Eq. (8b)]. Note that ${\widehat{G}}_{int}$ includes scattering by impurities only through $\widehat{\Sigma }\left\{G,V\right\}$, which is a functional of the propagators $\widehat{G}$ and $\widehat{V}$. The bare Green’s function, denoted by ${\widehat{G}}_b$, is free from the interactions and the scattering by impurities.

Figure 2

The Keldysh contour. The first, vertical, part of the contour is parallel to the imaginary axis starting at $t=-\infty +i\beta$ and ending at $t=-\infty +i\delta$. The second, horizontal, part of the contour is parallel to the real time axis. Here $\beta$ is the inverse temperature.

Figure 3

(a) The heat current vertex for noninteracting electrons. (b) The heat current vertex in the presence of interactions is a product of the renormalized velocity multiplied by the frequency of the incoming and outgoing legs.

Figure 4

The diagrammatic representation of the electric conductivity in the Fermi-liquid theory.

Figure 5

The self-energy before averaging over the disorder. After averaging, it will be decorated with the diffusons.

Figure 6

The series describing the interaction propagator.

Figure 7

The terms with a horizontal interaction line after averaging over the disorder.

Figure 8

The diagrams with a vertical interaction line. The obvious counterpart of diagram (b) is not shown.

Figure 9

The drag diagrams after the averaging over the disorder.

Figure 10

The self-energy in the second-order expansion with respect to the interaction.

Figure 11

A contribution to the thermal conductivity calculated using the simplified Kubo formula given in Eq. (2).

Figure 12

The contributions to the self-energy of the quasiparticles relevant for the discussion of the diagram presented in Fig. 11.

Figure 13

The two contributions to the Coulomb drag corresponding to $\mathcal{U}=\left[n_P\left(-\omega \right)V_{\mathbf{F}}^{<}\left(\mathbf{q},\omega \right)+n_P\left(\omega \right)V_{\mathbf{F}}^{>}\left(\mathbf{q},\omega \right)\right]$. (a) The diagrammatic interpretation for $V_{\mathbf{F}}^{<}\left(\mathbf{q},\omega \right)$ and (b) the diagrammatic interpretation for $V_{\mathbf{F}}^{>}\left(\mathbf{q},\omega \right)$.

Figure 14

The analytical structure of the different contributions to the drag diagram for (a) the electric current and (b) the heat current as a response to either an electric field or a temperature gradient.

Figure 15

The diagrammatic representation of the contributions to the currents described in Eq. (D2). For simplicity we do not show here the dashed lines that indicate the scattering by the impurities.

Figure 16

The diagrammatic representation of the contributions to the currents written in Eqs. (D4a, D4b).