Time evolution of the Luttinger model with nonuniform temperature profile

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 $T\left(x\right)$. As a specific example we consider the case when $T\left(x\right)$ is equal to $T_L$ $\left(T_R\right)$ 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 $t\ge 0$. 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 $T\left(x\right)$. 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.

Figure 1

Interacting fermions. Plots of analytical results for the energy density $e(x,t)=v[E(x,t)-E_0]/J$ in an interval $[-\ell ,\ell ]$ around $x=0$ at times (a) $t=0$ and (b) $t>0$ for the Luttinger model with local and nonlocal interactions. The results in the local case are given by (11) for $V\left(x\right)=\pi v_F\delta \left(x\right)/2$ and in the nonlocal case by (17) for $V\left(x\right)=\pi v_F/[4acosh(\pi x/2a)]$ with $a=0.100\ell$ and $a=0.200\ell$, respectively. The coupling constant is $\lambda =0.6$, and the other parameters are $\beta =20,\epsilon =-0.01,\delta =0.06\ell ,t_0=\ell /v_F$, and $v_F=1$. The value of $\epsilon$ is small enough that $O\left({\epsilon }^2\right)$ corrections are negligible.

Figure 2

Interacting fermions. Plots of analytical results for the heat current $j(0,t)=J(0,t)/J$ through $x=0$ for the Luttinger model using the same parameters as in Fig. 1. Also included is the local case without the second term in (13) (black dotted line).

Figure 3

Noninteracting fermions. Plots of analytical results for the energy density $e(x,t)=v[E(x,t)-E_0]/J$ in an interval $[-\ell ,\ell ]$ around $x=0$ at times (a) $t=0$ and (b) $t>0$ for the noninteracting Luttinger model and for the $XX$ chain. The results for the former are given by (11) and (12) with $v_F$ instead of $v$. The $XX$ chain is considered close to half filling on a lattice with spacing $a_0=0.025\ell$ and $a_0=0.050\ell$, respectively. The other parameters are the same as in Fig. 1.

Figure 4

Noninteracting fermions. Plots of analytical results for the heat current $j(0,t)=J(0,t)/J$ through $x=0$ for the noninteracting Luttinger model and for the $XX$ chain using the same parameters as in Fig. 3.