Interaction effects on thermal transport in quantum wires

We develop a theory of thermal transport of weakly interacting electrons in quantum wires. Unlike higher-dimensional systems, a one-dimensional electron gas requires three-particle collisions for energy relaxation. The fastest relaxation is provided by the intrabranch scattering of comoving electrons which establishes a partially equilibrated form of the distribution function. The thermal conductance is governed by the slower interbranch processes which enable energy exchange between counterpropagating particles. We derive an analytic expression for the thermal conductance of interacting electrons valid for arbitrary relation between the wire length and electron thermalization length. We find that in sufficiently long wires the interaction-induced correction to the thermal conductance saturates to an interaction-independent value.

Figure 1

Schematic picture of a quantum wire of length $L$. Electrons in the left and right leads are described by Fermi distribution functions characterized by temperatures $T_{l\left(r\right)}$ and chemical potentials ${\mu }_{l\left(r\right)}$. Due to three-particle collisions electrons may backscatter and also exchange energy between the subsystems of warmer right movers and colder left movers.

Figure 2

(a) Intrabranch relaxation of comoving electrons that establishes the partially equilibrated form of the distribution function in Eq. (5). (b) Dominant interbranch three-particle process for the energy exchange ${\dot{Q}}^R$ between counterpropagating electrons that contributes to the thermal conductance correction. (c) Leading three-particle collision which results in a finite rate ${\dot{N}}^R\propto e^{-\mu /T}$ and thus a temperature-dependent correction to the conductance of a short wire (Ref.19). (d) Equilibration mechanism: multistep diffusion through the bottom of the band of an electron from the right to the left Fermi point accompanied by the excitation of many electron-hole pairs (Refs.20, 21, 22).

Figure 3

Interaction-induced correction to the thermal conductance of a clean quantum wire as a function of its length plotted for different values of temperature (from the bottom to the top curve) $T/\mu =0.05,0.1,0.15,0.2$. For $L\ll {\ell }_b$ the correction scales with $L$ and saturates to a constant value proportional to ${(T/\mu )}^2$ once ${\ell }_b\ll L$ in accordance with Eqs. (56) and (57).

Figure 4

Direct (a) and five exchange (b)–(f) terms in the three-particle amplitude ${\mathcal{A}}_{123}^{1^{\prime }{2}^{\prime }{3}^{\prime }}$ [Eq. (B2)] that contribute to the finite momentum ${\dot{P}}^R$ and energy ${\dot{E}}^R$ exchange rates between right and left movers.