# Time evolution of the Luttinger model with nonuniform temperature profile

## TL;DR

This paper analyzes the time evolution of a one-dimensional Luttinger model with a nonuniform temperature profile, deriving exact and approximate expressions for energy, heat current, and correlations, revealing universal and nonuniversal features.

## Contribution

It provides analytical formulas for the nonequilibrium dynamics of the Luttinger model with nonuniform temperature, including exact results for local interactions and first-order approximations for nonlocal interactions.

## Key findings

- Exact expressions for energy density and heat current involving Schwarzian derivative.
- Universal heat current in the steady state even with broken conformal invariance.
- Predicted short-time peaks and dispersion effects consistent with numerical simulations.

## Abstract

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 $T(x)$. For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in $\epsilon$. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on $T_{L,R}$ agrees with numerical results for the $XXZ$ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.