Emergent Thermalization Thresholds in Unitary Dynamics of Inhomogeneously Disordered Quantum Systems

TL;DR

This work addresses how inhomogeneous disorder in a unitary quantum system can yield emergent thermalization thresholds, by coupling a large weakly disordered chain to a smaller strongly disordered chain and tracking spin transport. The authors combine numerical simulations with a perturbative Fermi’s Golden Rule framework and extreme-value theory to identify three regimes of thermalization as a function of the small-chain disorder $W_B$: full self-averaging thermalization at weak $W_B$, a non-self-averaging, realization-dependent regime at intermediate $W_B$, and inhibited transport at strong $W_B$, with the non-self-averaging interval expanding with the size $L_A$ of the large chain. A key analytical result is that the threshold for inhibited transport, $W_B^{thr}$, grows with $L_A$ for fixed $L_B$, explained via an Irwin–Hall distribution for spectral-width bounds and an extreme-value calculation for minimal level spacings. Overall, the study demonstrates that finite-size, inhomogeneous disorder can engineer observable thermalization–localization behavior in closed quantum systems, guiding experimental probes in ultracold-atom setups and informing the interpretation of disorder-averaged diagnostics.

Abstract

Inspired by the avalanche scenario for many-body localization (MBL) instability, we reverse the conventional set-up and ask whether a large weakly-disordered chain can thermalize a smaller, strongly-disordered chain when the composite system evolves unitarily. Using transport as a dynamical probe, we identify three distinct thermalization regimes as a function of the disorder strength of the smaller chain: (i) complete thermalization with self-averaging at weak disorder, (ii) realization-dependent thermalization with strong sample-to-sample fluctuations at intermediate disorder, and (iii) absence of thermalization at strong disorder. We find that for a fixed length of the smaller chain, the non-self-averaging regime broadens with the size of the weakly-disordered chain, revealing a nuanced interplay between disorder and system size. These results highlight how inhomogeneous disorder can induce emergent thermalization thresholds in closed quantum systems, providing direct access to disorder regimes where thermalization or its absence can be reliably observed.

Figures (9)

• Figure 1: In panel (a), we schematically show our set-up, where the blue region denotes the larger chain with weak disorder, while the red region represents the smaller, strongly disordered chain. The Hamiltonian of the system is given in Eq. \ref{['eq:H']}. Panel (b) presents for a fixed value of $L_\mathrm{B}$, in the $(L_\mathrm{A}, W_\mathrm{B})$ plane the three regimes (I), (II), and (III), detailed in the text, which show distinctively different thermalization behavior. The right boundary of regime (II) shifts toward larger $W_\mathrm{B}$ values with increasing $L_\mathrm{A}$, indicating the growth of this regime with the size of the weakly-disordered chain.
• Figure 2: For the system \ref{['eq:H']}, (a) and (b) show respectively the disorder-average and the most probable value of the magnetization $\widetilde{m}_\mathrm{B}$ of system B of size $L_\mathrm{B}=4$, with system A having sizes $L_\mathrm{A}=6$ -- $14$ in steps of $2$. Panels (c)--(h) depict the probability distribution of $\widetilde{m}_\mathrm{B}$ for disorder strengths $W_\mathrm{B}=0.15, 1.05, 1.55, 2.3, 6.05, 12.2$, and $L_\mathrm{A}=10$. Panel (i) shows variation of $\mu_4\equiv \overline{(\widetilde{{m}}_\mathrm{B}-\overline{\widetilde{{m}}}_\mathrm{B})^4}$ with $W_\mathrm{B}$ for various $L_\mathrm{A}$'s as in panel (a). We define $W_\mathrm{B}^\mathrm{ns}$ (indicated in panel (i) for $L_\mathrm{A} = 6$) as the range of $W_\mathrm{B}$ values over which $\mu_4$ takes up values above $0.3 \times 10^{-4}$ (horizontal dashed line in (i)). This is different from $W_\mathrm{B}^\mathrm{thr}$ (also shown in panel (i) for $L_\mathrm{A} = 6$), which is the value of $W_\mathrm{B}$ after which $\mu_4$ takes up values below $0.3 \times 10^{-4}$. Panel (j) shows $W_\mathrm{B}^{\mathrm{ns}}$ as a function of $L_\mathrm{A}$. The inset in (i) shows the tail behavior, depicting distinct differences for different $L_\mathrm{A}$'s, while that in (j) shows as a function of $L_\mathrm{A}$ the value $W_\mathrm{B}^\mathrm{thr}$ of $W_\mathrm{B}$ beyond which one has inhibited spin transport from A to B. Every panel corresponds to $J=1$, $W_\mathrm{A}=0.1$ and $5\times 10^4$ disorder realizations.
• Figure 3: For $L_\mathrm{A} = 11$ and $W_\mathrm{A} = 0.5$, panel (a) shows the eigenstates of $H_{L_\mathrm{A}}$ (red) with the states belonging to the $S^z = 0$ sector (black). Inset is a blow up of the middle part of the spectrum; Panel (b) on the other hand shows the distribution $P(\omega_{\beta \alpha})$ for 10 disorder realizations $\{ h_i^\mathrm{A} \}$. The inset shows the distribution of $\tilde{\omega}_{\beta \alpha}\equiv \overline{\omega_{\beta \alpha}^2} - (\overline{\omega_{\beta \alpha}})^2$ calculated for $1000$ realizations of $\{h_i^\mathrm{A} \}$. Panel (c) shows $\tilde{\omega}_{\beta \alpha}$ as a function of $L_\mathrm{A}$, with $L_\mathrm{B}=1$. We have set $J=1$.
• Figure 4: $L_\mathrm{B} = 1$: (a) $P(W_\mathrm{B}>\widetilde{\Delta})$ versus $W_\mathrm{B}$ for various $L_\mathrm{A}$'s. Panel (b) shows as a function of $L_\mathrm{A}$ the threshold $\tilde{W}_\mathrm{B}^\mathrm{thr}$ above which $P(W_\mathrm{B}>\widetilde{\Delta})> 0$ (here chosen to be $0.5$). $L_\mathrm{B} = 4$: (c) $P(\mathbb{E}[\tilde{h}_\mathrm{min}]>\tilde{\Delta})$ versus $W_\mathrm{B}$ for various $L_\mathrm{A}$'s. Panel (d) shows as a function of $L_\mathrm{A}$ the threshold $\tilde{W}_\mathrm{B}^\mathrm{thr}$ above which $P(\mathbb{E}[\tilde{h}_\mathrm{min}]>\tilde{\Delta})> 0$. For all panels, $J=1$, $W_\mathrm{A}=0.1$.
• Figure 5: The different panels in the figure show the same quantities as in Fig. \ref{['fig:fig_2']} , but with $h_i^\mathrm{A}$, and $h_j^\mathrm{B}$ sampled from Gaussian distributions $\mathcal{N}(0, W_\mathrm{A})$, and $\mathcal{N}(0, W_\mathrm{B})$, respectively. We fix $W_\mathrm{A}=0.1$. The figure shows that the information conveyed by Fig. \ref{['fig:fig_2']} also holds when the choice of distribution for the fields $h_j^\mathrm{B}$ is changed from uniform to Gaussian.