Thermalization of Nonequilibrium Electrons in Quantum Wires

We study the problem of energy relaxation in a one-dimensional electron system. The leading thermalization mechanism is due to three-particle collisions. We show that for the case of spinless electrons in a single channel quantum wire the corresponding collision integral can be transformed into an exactly solvable problem. The latter is known as the Schrödinger equation for a quantum particle moving in a Pöschl-Teller potential. The spectrum for the resulting eigenvalue problem allows for bound-state solutions, which can be identified with the zero modes of the collision integral, and a continuum of propagating modes, which are separated by a gap from the bound states. The inverse gap gives the time scale at which counterpropagating electrons thermalize.

Figure 1

Three-particle scattering processes that allow for the energy exchange between counterpropagating electrons and thus lead to their thermalization. The matrix elements $⟨1^{\prime }{2}^{\prime }{3}^{\prime }|\widehat{V}\widehat{G}\widehat{V}|123{⟩}_c$ decompose into six contributions, $A({11}^{\prime },{22}^{\prime },{33}^{\prime })$ plus five permutations of primed arguments, corresponding to the direct (a) and five exchange terms (b)–(f).