Time evolution of the Luttinger model with nonuniform temperature profile

Understand

We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile $T(x)$.

• As a specific example we consider the case when $T(x)$ is equal to $T_L$ ($T_R$) far to the left (right).
• Using a series expansion in $\epsilon = 2(T_{R} - T_{L})/(T_{L}+T_{R})$, we compute the energy density, the heat current density, and the fermion two-point correlation function for all times $t \geq 0$.
• For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of $T(x)$.