Upper and Lower Bounds for The Quantum Dynamics of One-Dimensional Divergence-Type Random Jacobi Operators

TL;DR

This work analyzes quantum transport for a one-dimensional divergence-gradient random Jacobi operator with bounded, positive i.i.d. coefficients. By combining the asymptotics of the integrated density of states and the Lyapunov exponent near the critical energy $E=0$ with large-deviation estimates for transfer matrices via the Figotin–Pastur phase formalism, the authors derive both upper and lower power-law bounds on time-averaged $q$-moments $M_T^q$. They establish nontrivial transport exponents: $\beta^-_q \ge \tfrac{1}{2}-\tfrac{2}{q}$ for $q\ge4$, and $\beta^+_q \le 1-\tfrac{1}{5q}$, along with an almost-sure lower bound for $q\ge\tfrac{11}{2}$ with exponent at least $\tfrac{2}{5}-\tfrac{11}{5q}$. The analysis relies on a detailed understanding of the IDS near $E=0$, the linear behavior of $L(E)$, and a robust bootstrap LDT for transfer matrices, revealing dynamics that appear nearly diffusive in numerics and highlighting the delicate interplay between spectral localization and anomalous transport in this div-grad setting.

Abstract

We study quantum transport for the discrete one-dimensional random Jacobi operator of divergence-gradient type. For strictly positive and bounded random variables, we analyze the q-moments of the position operator and establish both upper and lower power-law bounds on their growth. Our approach relies on the asymptotic behavior of the integrated density of states and the Lyapunov exponent near the critical energy 0, previously obtained by Pastur and Figotin. A key ingredient in our analysis is the large deviation-type estimates explored via the phase formalism, which play a central role in deriving bounds on the growth of the transfer matrices.

Figures (9)

• Figure 1: Log–log plots of the non-averaged $q$-th moment $M^q(t) = \sum_{k=0}^{n-1} k^q \,|\langle \delta_k, e^{-it H_{n,\omega}} \delta_0 \rangle|^2,$ where $H_{n,\omega}$ is the restriction of the random operator $H_{\omega}$ to $\{0,1,\dots,n-1\}$. The random coefficients satisfy $a_i \sim \text{Uniform}[1,2]$. System size: $n = 10{,}000$. Panels correspond to $q =1, 2, 4, 6, 10, 60$ (top-left to bottom-right). Red solid line: numerical data; blue dashed: reference slope $\max(0, q - \tfrac{1}{5})$; green dashed: reference slope $\max(0, \tfrac{q}{2} - 2)$; purple dashed: reference slope $\tfrac{q}{2}$. Time grid: $t \in \{ 100, 150, \cdots, 10000 \}$.
• Figure 2: (Normalized) Eigenfunction plots $\psi(i)$ for the div-grad model $H_\omega$ with random entries $a_j \sim \text{Uniform}[1,2]$, restricted to a finite size $n = 3000$. The first panel corresponds to the ground state near $E_0 = 0$. Panels 2--4 show eigenfunctions near target energies $E_1 = 1/(kn)$, $E_2 = 2/(kn)$, and $E_3 = 20/(kn)$, where the localization length $\ell \approx 1/(kE)$ is roughly $n$, $n/2$, and $n/20$, respectively. Here $k$ is the linear constant in the asymptotic formula \ref{['eqn:linearLE']}, computed explicitly from $\kappa$ and $a_j$. As $E$ decreases to $0$, the eigenfunctions become less localized, illustrating the growth of the localization length predicted by the Lyapunov exponent in \ref{['eqn:linearLE']}.
• Figure 3: Finite volume IDS $N_n(E)$ with $n=3000$ for two cases: (left) i.i.d. $a_i \sim \text{Uniform}[0.1,1]$ and (right) i.i.d. $a_i \sim \text{Bernoulli}\{0.1,1\},\mathbb{P}(0.1)=0.5$. The plots show the integrated density of states with reference curves $y=\frac{\sqrt E}{\pi\sqrt \kappa}$, with $\kappa$ given by \ref{['eqn:NE-root']}.
• Figure 4: Monte Carlo mean estimates of the Lyapunov exponent $L_n(E)=\frac{1}{n}\log \|T_n^E\|$ for $n = 3000$ under different uniform distributions of $a_j$. Top row: $a_j \sim \mathrm{Uniform}[0.1, 1]$, averaged over 100 replicates (samples). Bottom row: $a_j \sim \mathrm{Uniform}[0, 1]$, averaged over 100 replicates (samples). Each column shows: (a) linear scale with reference slope $kE$ from \ref{['eqn:linearLE']}, where $k = \frac{\kappa }{8} \, \mathbb{E} \{ ( 1/a_j - 1/\kappa )^2 \}$ and (b) log--log scale with slope 1 (blue, dashed) and slope $\tfrac{1}{2}$ (green, dot-dashed) reference lines. Red curves represent numerical estimates. Note: When $\inf \operatorname{supp} P_0 = 0$, we have $\kappa = 0$ in \ref{['eqn:linearLE']}.
• Figure 5: Visualization of the regions contributing to the bounds in Lemma \ref{['lem:CT-logT']}. The green shaded area corresponds to energies far from the spectrum ($|E|\ge E_1$) or large frequencies ($n \ge T^{1+\alpha}$), where the Combes--Thomas estimate ensures exponential decay of the Green's function. The blue shaded area represents energies near the spectrum but away from the critical point $E = 0$, with frequencies up to $T^{1+\alpha}$; the dashed line at $n = (\log T)^3$ indicates the scale relevant for the logarithmic bound in \ref{['eqn:Mtq-logT']}. The red segment on the $E$-axis marks the spectrum of the operator as given in \ref{['eqn:spe']}, with its lower endpoint at $E = 0$, the only critical energy where the Lyapunov exponent vanishes, and its upper endpoint at $E_{\max} = 4a_+$.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: pastur1992book
• Remark 1.1
• Remark 1.2
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Proposition 2.2
• proof