Quantum typicality approach to energy flow between two spin-chain domains at different temperatures

TL;DR

This work extends dynamical quantum typicality (DQT) to study energy transport between two spin-chain domains at different temperatures, including low-temperature dynamics. By simulating bipartite setups across XX, critical Ising, and XXZ chains and benchmarking against universal CFT results and generalized hydrodynamics, the authors show that the non-equilibrium steady-state currents agree with theoretical predictions, such as $J_E = (\pi c/12)(T_L^2 - T_R^2)$ and bottleneck forms involving $\min(c_L,c_R)$. The results demonstrate DQT's reliability and efficiency for closed quantum systems with interfaces, capturing both global currents and local densities/contacts, and revealing transport features like bottlenecks and finite-size effects. The study thus validates DQT as a scalable tool for probing bipartite energy transport at low temperatures and lays groundwork for extending to other critical or gapped models.

Abstract

We discuss a quantum typicality approach to examine systems composed of two subsystems at different temperatures. While dynamical quantum typicality is usually used to simulate high-temperature dynamics, we also investigate low-temperature dynamics using the method. To test our method, we investigate the energy current between subsystems at different temperatures in various paradigmatic spin-1/2 chains, specifically the XX chain, the critical transverse-field Ising chain, and the XXZ chain. We compare our numerics to existing analytical results and find a convincing agreement for the energy current in the steady state for all considered models and temperatures.

Figures (7)

• Figure 1: Schematic representation of the setup. The total system consists of two subsystems described by $H_{L}$ and $H_{R}$, respectively. Initially, they are prepared at temperatures $T_L$ and $T_R$. The interaction of both parts for times $tJ \geq 0$ is described by the Hamiltonian $H_C$.
• Figure 2: Time evolution of the energy current for temperatures $T_{L}/J=1/6, T_{R}/J=1/8$, total system size $L=28$ for the coupling of (a) two XX chains, (b) two critical Ising chains and (c) one XX chain to a critical Ising chain. Numerical data for two different pure random initial states ($N$=1) and for the average over many initial states (N=100) are compared to the CFT value. Note that larger fluctuations at longer times do not contradict the bound in Eq. (\ref{['eq:bound']}), since this bound is not tight.
• Figure 3: Similar data as the one in Fig. \ref{['fig:time_evo']} but now for different total system sizes $L$. Here, the numerical data are again averaged over $N = 100$ pure random initial states. For larger system sizes $L$, finite size effects are visible for later times $tJ$.
• Figure 4: Steady-state energy current $J_{0}$ versus temperature of the right subsystem $T_{R}$ for the coupling of (a) two XX chains, (b) two critical Ising chains, and (c) one XX chain to a critical Ising chain for the total system size $L=24$ and $T_{L}/J = 1/8$. The numerical data are compared to the CFT result (\ref{['eq:energy_cft']}).
• Figure 5: Steady-state profile for (a) the local density $\langle h_{r}\rangle$ and (b) the local current $\langle J_{r}\rangle$ for the coupling of two XXZ chains with the same anisotropies $\Delta$, $T_{L}/J=1$,$T_{R}/J=1/2$, and total system size $L=24$. (Note that in the $x$-axis the position $r$ is divided by the time $t$.) The numerical data are compared to the results from GHD in Ref. Bertini2016. Here, the numerical data are shown for both, a random product state (open symbols) and a random state in the entire Hilbert space (closed symbols).