Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model

TL;DR

The paper rigorously demonstrates normal typicality and dynamical typicality for a structured random block-band Hamiltonian with block-dependent variances, enabling tunable intermediate equilibration times via a small parameter $\lambda$. By leveraging ETH for Wigner-type matrices, two-resolvent local laws, contour-integral representations, and stationary-phase analysis, the authors derive precise asymptotics for $\mathcal{M}_{\mu\nu}$ and $w_{\mu\nu}(t)$ in both two- and three-block models, with explicit $t^{-3}$ corrections whose sign and magnitude depend on block geometry. The results show that equilibration times interpolate between extreme fast and slow regimes, controlled by the relative macro-space sizes, and extend to multi-macro-space settings with analogous structure. Overall, the work provides a rigorous, quantitative framework for intermediate equilibration in structured random quantum systems and introduces tools likely applicable to a broader class of inhomogeneous random matrices.

Abstract

We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices [arXiv:2403.10359] and an intricate analysis of the deterministic approximation.

Figures (2)

• Figure 1: Numerical simulation of the functions $t\mapsto \|P_\nu\psi_t\|^2$ for a random $2\times 2$ block matrix as in Assumption \ref{['ass:2']}. Here, $\lambda=0.2$ and the Hilbert space $\mathcal{H}$ of dimension $N=4200$ is decomposed into 2 macro spaces of dimensions $d_1 = 700$ (green curve) and $d_2=3500$ (blue curve). The initial state $\psi_0\in\mathbb{S}(\mathcal{H}_1)$ was chosen purely randomly. The black solid curves are the deterministic (w.r.t. the randomly chosen $\psi_0$) approximations $w_{1\nu}(t)$ from \ref{['eq:wmunus']} and the black dashed curves are the approximations of $\|P_\nu\psi_t\|^2$ according to Theorem \ref{['thm: Pnupsi']}. Note that the dashed curves start from $t\geq 1/(2\lambda) = 2.5$ as the approximations in Theorem \ref{['thm: Pnupsi']} are only meaningful for $t \gg 1/\lambda$.
• Figure 2: Numerical simulation of the functions $t\mapsto\|P_\nu\psi_t\|^2$ for a random $3\times 3$ block matrix as in Assumption \ref{['ass:3']}. Here, $\lambda=0.2$ and the Hilbert space $\mathcal{H}$ of dimension $N=4340$ is decomposed into 3 macro spaces of dimensions $d_1 = 140$ (green curve), $d_2=700$ (blue curve) and $d_3=3500$ (red curve). As in Figure \ref{['fig:2blocks']}, the initial state $\psi_0\in\mathbb{S}(\mathcal{H}_1)$ was chosen purely randomly. The black solid curves are the deterministic (w.r.t. the randomly chosen $\psi_0$) approximations $w_{1\nu}(t)$ and the black dashed curves are the approximations of $\|P_\nu\psi_t\|^2$ according to Theorem \ref{['thm: Pnupsi3']}. Again, the dashed curves only start from $t\geq 1/(2 \lambda) = 2.5$. A similar figure appeared as a purely numerical experiment in TTV23aTTV23b in the case of a random band matrix and four macro spaces. Here, although in a slightly different model, we rigorously prove the same qualitative behavior of the $t \mapsto \|P_\nu \psi_t\|$ curves.

Theorems & Definitions (18)

• Definition 1.1: Stochastic domination
• Theorem 2.2: Normal typicality: $\mathcal{M}_{\mu \nu} \approx d_\nu/N$
• Theorem 2.3: Dynamical typicality: The $w_{\mu \nu}$'s
• Remark 2.4: Explicit form of the $w_{\mu \nu}$'s
• Theorem 2.5: Approach to equilibrium
• Theorem 2.7: Normal typicality: $\mathcal{M}_{\mu \nu} \approx d_\nu/N$
• Theorem 2.8: Dynamical typicality: The $w_{\mu \nu}'s$
• Remark 2.9: Explicit form of the $w_{\mu \nu}$'s
• Theorem 2.10: Approach to equilibrium
• Lemma 3.1: Relations between the $w_{\mu\nu}$