Quantum thermalization and the route to ergodicity

TL;DR

The paper analyzes quantum thermalization in coupled chaotic subsystems using a bipartite Bose-Hubbard model with $H=H_0+λV$, revealing a decoupling between spectral chaos and ETH-type ergodicity. It introduces and exploits the energy-shell concept, showing a robust, largely λ-independent shell width $Δx_{ ext{shl}}$ alongside a linear-in-$λ$ second-moment spreading $Δε_{ ext{shl}} \\sim λ$. Five coupling regimes (Perturbative, Mixing, Wigner, Ergodic, Integrable) emerge, with distinct horizontal and vertical ergodization and clear regime borders that scale with system size, indicating a non-universal route to ergodicity. The study contrasts this with WBRM predictions, emphasizes the irrelevance of subsystem geometry, and highlights the need to distinguish chaos (spectral statistics) from ergodicity (ETH-type thermalization) in finite quantum systems.

Abstract

We consider a minimal model for quantum thermalization of coupled chaotic subsystems. The route towards ergodicity is explored as a function of the coupling strength. The results are contrasted with the predictions of standard Random Matrix Theory (RMT) and the Eigenstates Thermalization Hypothesis (ETH). We highlight a coupling regime of disparity between the spectral statistics that indicates chaos, and ergodicity measures that indicate lack of ETH thermalization. The analysis involves a revision of the energy shell concept, in a way that is consistent but independent of the semiclassical perspective.

Figures (5)

• Figure 1: The unperturbed basis. The unperturbed energies $\varepsilon_{x}$ and $\varepsilon_{x,n}$ for the WBRM (a) and for the BHM (b,c), respectively. The bandwidth $\Delta_b$ is the maximal range in energy that can be reached by a single transition from an initial state. Possible first-order transitions are indicated. The width of the energy-shell $\Delta x_{\text{shl}}$ is defined via the microcanonical distribution $p_x$ (d). It is not sensitive to $\lambda$ for the BHM, and has a well defined finite value in the limit of zero coupling. In contrast, the WBRM model can be regarded as a singular limit that features ${\Delta x_{\text{shl}} \propto \lambda}$.
• Figure 2: The width of the energy shell.(a) The width $\Delta x_{\text{shl}}$ versus the width $\delta E$ of the arbitrary energy window about ${E=0.41666}$ for $\lambda=10^{-3}$ (solid blue line), $10^{-2}$ (dashed red), $10^{-1}$ (dotted orange), and $1.0$ (dash-dotted purple). The width of the energy shell is well defined (assuming quantum mechanically large but classically small $\delta E$) and the dependence of the obtained shell width on the coupling strength $\lambda$ is weak. (b,c) The dependence of $\Delta x_{\rm shl}$, ${\cal M}_{\rm shl}$ and $\Delta \varepsilon_{\rm shl}$ on the coupling $\lambda$. In the latter case one observe a residual value for small $\lambda$, corresponding to the arbitrary $\delta E$ (dashed line is for $\delta E=0.0333$ whereas dash-dotted line is for $\delta E=0.0166$). This numerical artifact can be eliminated by looking on ${\overline{\Delta \varepsilon_{\nu}}}$ (solid line), even if the eigenstates are non-ergodic! The dotted line is a guide to the eye, corresponding to linear $\Delta\varepsilon_{\rm shl}(\lambda)$ dependence
• Figure 3: Chaos and Ergodicity Measures. Misc measures are calculated for the energy shell ${0.4<E<0.4333}$ as a function of the coupling strength $\lambda$, and normalized per the expected GOE values. The spectral measure for chaos $\tilde{r}$ (square markers) is correlated with $\overline{\Delta x_{\nu}}/\Delta x_{\rm shl}$ (solid line). The ETH measure $\sigma_x /\Delta x_{\rm shl}$ (dotted line) is an insufficient measure for ergodization. The measure $\overline{\mathcal{M}_{\nu}}/(\mathcal{M}_{\rm shl}/3)$ (dashed line) is an effective measure for the actual identification of the ergodic regime. We also plot the mean peak probability $\overline{P_{\nu} }$ (dash-dotted line), which is used to identify the first-order perturbative regime (arbitrarily defined as $\overline{P_{\nu} }>0.6$). Inverted triangles mark the coupling strengths for the states presented Fig.\ref{['fg4']}.
• Figure 4: Representative eigenstates.Upper row: Images of $p^{(\nu)}_{x,n}$ for a representative eigenstate with ${E\approx 0.425}$. Each panel is for a different coupling corresponding (from left to right) to the Perturbative, Mixing, Wigner, and Ergodic regimes. Second row: The resulting imbalance distribution $p_x^{(\nu)}$. Third row: The $p_{x,n}^{(\nu)}$ probabilities are plotted versus $(x,n)$, arranged in order of ascending unperturbed energy. Bottom row: The same $p_{x,n}^{(\nu)}$ probabilities are compared on a log scale to the coarse-grained 1st-order perturbation profiles (solid lines). White lines are coarse-graining of $p_{x,n}^{(\nu)}$.
• Figure 5: Chaos and Ergodicity Measures. Same as Fig.\ref{['fg3']} but with random internal couplings between all the sites of the coupled subsystems, selected from a normal distribution with zero mean and unit standard deviation. Thus, each subsystem is not a chain, but a fully connected cluster of arbitrary geometry.