On the difference between thermalization in open and isolated quantum systems: a case study

TL;DR

The paper resolves an apparent dichotomy between open and isolated quantum system thermalization by showing that the same DQD+lead model exhibits both OQS and IQS thermalization, with the key difference being the order of the thermodynamic and long-time limits. Using dynamical typicality, the authors bridge the OQS and IQS frameworks in a single dynamical setup and identify a crossover time $t_{\rm oqs}$ that scales with bath size as $t_{\rm oqs} \propto L_B/g_B$, separating fast OQS relaxation to the mean-force Gibbs state from slow IQS relaxation that emerges when the bath is effectively finite. The interacting DQD ($V>0$) shows strong evidence of IQS thermalization consistent with ETH-like behavior, while the free case ($V=0$) remains non-thermal in the IQS sense; nonetheless, OQS thermalization occurs in both cases. These results clarify how limit-order effects shape thermalization and offer a controlled framework for comparing OQS and IQS perspectives in quantum impurity models. The findings have implications for understanding relaxation in nanoscale and impurity systems and for exploring thermalization mechanisms beyond weak-coupling assumptions.

Abstract

Thermalization of isolated and open quantum systems has been studied extensively. However, being the subject of investigation by different scientific communities and being analysed using different mathematical tools, the connection between the isolated (IQS) and open (OQS) approaches to thermalization has remained opaque. Here we demonstrate that the fundamental difference between the two paradigms is the order in which the long time and the thermodynamic limits are taken. This difference implies that they describe physics on widely different time and length scales. Our analysis is carried out numerically for the case of a double quantum dot (DQD) coupled to a fermionic lead, also known as the interacting resonant level model in quantum impurity physics. We show how both OQS and IQS thermalization can be explored in this model on equal footing, allowing a fair comparison between the two. We find that while the quadratically coupled (free) DQD experiences no isolated thermalization, it of course does experience open thermalization. For the non-linearly interacting DQD coupled to a fermionic lead, the many-body interaction in the DQD breaks the integrability of the whole system. We find that this system shows strong evidence of both OQS and IQS thermalization in the same dynamics, but at widely different time scales, consistent with reversing the order of the long time and the thermodynamic limits.

Figures (8)

• Figure 1: Schematic of DQD coupled to a fermionic lead. The system of interest is that of two fermionic sites (blue discs) which form a double quantum dot (DQD). The DQD is linearly coupled to a chain of $L_B$ fermionic sites (yellow discs) that represent a fermionic lead (bath). The hopping (blue lines) within the lead is $g_B$. The hopping constant within the DQD is $g$, while between the DQD and the lead it is $\gamma$. In addition, a nearest neighbour repulsive many-body interaction with strength $V$ (grey arrow) may act between the two sites of the DQD. The Hamiltonian for the full, closed, spinless fermion chain (orange box) is given by \ref{['eq:Htot']}. We refer to $V=0$ as the free fermion model, and to $V>0$ as the interacting DQD model.
• Figure 2: Crossover between OQS and IQS regimes: Shown is the dynamics of a representative DQD observable, $\langle \hat{n}_1 (t) \hat{n}_2 (t) \rangle$, for different bath sizes $L_B$, starting from an initial state of the form in Eq. \ref{['typical_initial_state']}, with two different sets of randomly chosen parameters in $\hat{H}_S^{\rm ini}$, see Eq. \ref{['H_S_ini']}, corresponding to two randomly chosen initial reduced states for the DQD. The left panels, (a) and (b), show the dynamics for the free model ($V=0$) for the two different initial condition of the DQD. Panels (c) and (d) show the dynamics for the interacting model ($V=g$), for the same two initial condition of the DQD as above. The horizontal dashed lines show the expectation values of the mean force Gibbs state $\hat{\rho}_{\rm MGS}$ corresponding to the cases $V=0$ and $V=g$, respectively. The $x$-axis is shown on log-scale to capture features over a wide time regime. The shaded region corresponds to the time up to which plots for various $L_B$ overlap, which is the OQS regime. In the remaining time regime, which we denote the IQS regime, the DQD feels the finiteness of $L_B$. The data in the OQS regime shows strong evidence of OQS thermalization for both $V=0$ and $V=g$. The data in the IQS regime shows strong evidence of IQS thermalization for $V=g$, as we discuss in the following sections in the main text. Other parameters: $g_B=2g$, $\gamma=g$, $\beta g=0.1$.
• Figure 3: Thermalization in the OQS regime: Panels (a), (b), (c) are for the free fermion case, $V=0$. Panels (d), (e), (f) are for the interacting DQD case with $V=g$. The results in (a), (d) corresponds to the same simulation as in Fig. \ref{['fig:OQS_IQS_crossover']}(a) and (c), now plotted only for $t \leq t_{\rm oqs}$, which corresponds to the OQS regime. Note that for each $L_B$, the time regime $t \leq t_{\rm oqs}$ is different. Plots for larger values of $L_B$ overlap with those of smaller values of $L_B$ but extend to longer times. In panels (b) and (e), we fix $L_B=26$ and consider two randomly chosen initial states (blue and yellow symbols) of the DQD. They correspond to the same two initial conditions chosen in Fig. \ref{['fig:OQS_IQS_crossover']}. The horizontal dashed lines in panels (a), (b), (d), (e) show the corresponding expectation value obtained with the $\hat{\rho}_{\rm MGS}$. In panels (c) and (f), we plot the trace distance between $\hat{\rho}_S(t_{\rm oqs}; L_B)$ and $\hat{\rho}_{\rm MGS}$ as a function of bath size $L_B$ for the two different choices of initial states (blue and orange symbols) of the DQD. The straight lines (dashed-dotted blue and dashed red) are exponential fits to the symbols, with numerical fit parameters given. Other parameters: $g_B=2g$, $\gamma=g$, $\beta g=0.1$, $L_0=8$.
• Figure 4: Thermalization in the IQS regime: Plot of the same data as in Fig. \ref{['fig:OQS_IQS_crossover']}(a) and (c), only with the $x$-axis in linear scale instead of log-scale. In contrast to Fig. \ref{['fig:OQS']}, where we considered maximal times up to $gt_{\rm oqs}\sim 5-8$, we now consider the whole time range of $gt$ up to $2000$. The horizontal dashed lines in panels (a) and (b) show the corresponding expectation value obtained with $\hat{\rho}_{\rm MGS}$. Panel (a) shows the free, integrable, DQD case $V=0$. Oscillations are found to persist for all times and for all the chosen values of $L_B$. Panel (b) shows the dynamics in the interacting, non-integrable$^*$ case $V=g$. The large oscillations disappear after some time $gt\sim 400$ and only small fluctuations about a relaxed value persist. Those small fluctuations diminish rapidly with increasing bath size $L_B$. Other parameters: $\beta g=0.1$, $\gamma=g$, $g_B=2g$.
• Figure 5: IQS thermalization of interacting DQD with variable bath size $L_B$: (a) Log-plot of the variance $\overline{\delta\langle \hat{n}_1\hat{n}_2 \rangle^2} (t_1,t_f)$ of data points shown in Fig. \ref{['fig:ETH_thermalization0']}(b), as a function of initial interval time $t_1$, while keeping the final interval time fixed, $gt_f=2000$. Here $t_1$ is varied between $0\leq t_1 \leq 1900g^{-1}$. Different colours indicate results for various bath sizes $L_B$. After a time $t^*(L_B)$, the dynamics reaches an approximately constant value which we call $v(L_B)$ (horizontal dashed lines). The spacing between these saturation values is equal as $L_B$ varies over the values $L_B = 20, 22, 24, 26$, which indicates exponential decay of $v(L_B)$ with $L_B$. (b) Log-plot of the long-time variance $\overline{\delta\langle \hat{A} \rangle^2}(t_1,t_f)$ for each of the four observables $\hat{A}$ of the interacting DQD, as a function of $L_B$. The times $t_1$, $t_f$ are taken large, so that the variance has reached the approximately constant value (c.f. panel (a)). The quantities $\overline{\delta\langle \hat{A} \rangle^2}(t_1,t_f)$ are numerically evaluated via Eq. \ref{['def_time_variance']} with $t_1=1800g^{-1}$ and $t_f=2000g^{-1}$. The plots show that the variance of each observable $\hat{A}$ decays exponentially with $L_B$, in line with Eq. \ref{['eq:expdecaywLB']}. Other parameters: $V = g, \beta g=0.1$, $\gamma=g$, $g_B=2g$.