One of the most straightforward ways to study thermal properties beyond linear response is to monitor the relaxation of an arbitrarily large left-right temperature gradient $T_L-T_R$. In one-dimensional systems which support ballistic thermal transport, the local energy currents $langle j(t)rangle$ acquire a non-zero value at long times, and it was recently investigated whether or not this steady state fulfills a simple additive relation $langle j(ttoinfty)rangle=f(T_L)-f(T_R)$ in integrable models. In this paper, we probe the non-equilibrium dynamics of the Hubbard chain using density matrix renormalization group (DMRG) numerics. We show that the above form provides an effective description of thermal transport in this model; violations are below the finite-time accuracy of the DMRG. As a second setup, we study how an initially equilibrated system radiates into different non-thermal states (such as the vacuum).
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