Nonequilibrium thermal transport and vacuum expansion in the Hubbard model

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 $\langle j\left(t\right)\rangle$ acquire a nonzero value at long times, and it was recently investigated whether or not this steady state fulfills a simple additive relation $\langle j(t\to \infty )\rangle =f\left(T_L\right)-f\left(T_R\right)$ in integrable models. In this paper, we probe the nonequilibrium 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 nonthermal states (such as the vacuum).

Figure 1

Main setups considered in this paper. A Fermi-Hubbard chain is prepared in thermal equilibrium at a temperature $T_L$. At time $t=0$, it is connected to a vacuum state or to a chain exhibiting a different temperature $T_R$. The former describes the expansion after switching off a (sharp) potential barrier; the latter models the evolution of an initial temperature gradient.

Figure 2

Thermal current flowing through the Hubbard chain in the presence of an initial temperature gradient defined by ${\beta }_{L,R}=1/T_{L,R}$. The on-site and nearest-neighbor interaction strength are $U=4$ and $V\in \{0,U/8\}$, respectively. In the integrable case $V=0$, the currents saturate to a nonzero stationary-state value.

Figure 3

Thermal current for the integrable model with fixed ${\beta }_R\in \{0,1,2\}$ and various ${\beta }_L$ (increasing from top to bottom). The on-site interaction strength is given by $U=1$ in (a),(b) and $U=4$ in (c),(d) (the charge gap is of size $\sim 0.04$ and $\sim 1.17$, respectively). The inset to (d) shows the data for ${\beta }_L=0.2$ on a magnified scale. If the curves for the different ${\beta }_R$ are shifted vertically (by the same amount for all ${\beta }_L$), they seem to converge to the same long-time asymptote, which thus depends only on ${\beta }_L$. This illustrates that on the time scales accessible by the DMRG, the stationary state is effectively described by the additivity relation of Eq. (1); violations are smaller than the resolution of the method.

Figure 4

Temperature dependence of the function $f\left(T\right)$ which governs the steady-state current via Eq. (1).

Figure 5

Same as in Fig. 3, but comparing the currents flowing into a thermal state at ${\beta }_R=1$ with an expansion into the vacuum for various ${\beta }_L$ of the left system. The on-site interaction is (a) $U=1$ and (b) $U=4$. [The curve for ${\beta }_L=3$ is not shown in (b).]

Figure 6

Full spatial profiles of the energy density $\langle h_n\left(t\right)\rangle$ and the energy current $\langle j_n\left(t\right)\rangle$ in the FHM at $U=4$ for the expansion of a thermal state with ${\beta }_L\in \{0,2\}$ into the vacuum for two times $t$.

Figure 7

Thermal current flowing in a XXZ spin chain with $\mathrm{\Delta }=0.5$ (this value governs the time evolution) if a thermal state with an inverse temperature ${\beta }_L$ on the left is connected to different states on the right: (i) a thermal state of the same chain, (ii) the vacuum, (iii) a half-polarized Néel state, and (iv) a thermal state of a XXZ chain with a different $\mathrm{\Delta }=1$. The curves for (ii)–(iv) are shifted vertically (by the same amount for all ${\beta }_L$). Note that data for (iii) and (iv) could only be calculated up to $t\approx 8$ due to the fast growth of the bond dimension for these global quenches.

Figure 8

Same as in Fig. 7 but in the gapped phase with $\mathrm{\Delta }=3$ and for various thermal states on the right. The curves for ${\beta }_R=0$ and ${\beta }_R=2$ were shifted vertically.