Phase-Averaged Delocalization Bounds

• Phase-Averaged Delocalization Bounds are defined as mathematical tools that quantify quantum transport by averaging quantum amplitudes over phase variables in complex systems.
• They leverage spectral analysis and the concept of eventual absolute continuity to establish sharp lower and upper bounds, indicating near-ballistic transport under certain conditions.
• This framework extends to random band matrices via fluctuation-averaging, effectively bridging deterministic and stochastic approaches to quantum delocalization.

Phase-averaged delocalization bounds quantify quantum transport phenomena in systems exhibiting deterministic or random structure, focusing on the averaged behavior over phase variables of underlying Hamiltonians. In recent work, sharp phase-averaged bounds were established for the Fibonacci Hamiltonian using spectral analysis, transport exponents, and a new notion of "eventual absolute continuity," substantially strengthening prior results on quantum delocalization in quasiperiodic and random models (Leclerc, 7 Dec 2025, Yang et al., 2018).

1. Phase-Averaged Transport Quantities

Given a self-adjoint operator $H_\omega$ (e.g., the Fibonacci Hamiltonian) with coupling $V>0$ and phase $\omega\in\mathbb T$, one studies the time evolution in position space using

$$a(n, t; \omega) = \langle \delta_n, e^{-iH_\omega t} \delta_0 \rangle,$$

interpreting $|a(n, t; \omega)|^2$ as the quantum probability amplitude from $0$ to $n$ at time $t$. The phase-averaged $p$-th moment at time $T$ is

$$M_p(T) := \sum_{n\in\mathbb Z} |n|^p \left\langle |a(n, T; \omega)|^2 \right\rangle_\omega,$$

with phase average $$\langle \cdot \rangle_\omega = \int_\mathbb T (\cdot)\, d\omega$$. Transport exponents

$$\beta^+(p) := \limsup_{T \to \infty} \frac{\log M_p(T)}{p \log T}, \quad \beta^-(p) := \liminf_{T \to \infty} \frac{\log M_p(T)}{p \log T},$$

characterize the asymptotic spreading rate, distinguishing between ballistic ($\beta^\pm(p)\approx1$), sub-ballistic, or localized quantum transport.

2. Spectral Conditions: Eventual Absolute Continuity

The classical absolutely continuous (a.c.) subspace $\mathcal H_\mathrm{ac}$ for $H$ is defined by spectral measures $\mu_\psi$ having densities with respect to Lebesgue measure: $\mu_\psi \ll dE$. The stricter requirement of absolute continuity is relaxed in (Leclerc, 7 Dec 2025) through "eventual absolute continuity": a vector $\psi$ lies in the subspace $\mathcal H_\mathrm{eac}$ if there exists $N\ge1$ such that the $N$-fold convolution measure

$$\mu_\psi^{*N} = \underbrace{\mu_\psi * \cdots * \mu_\psi}_{N \text{ times}}$$

is absolutely continuous, i.e., $\mu_\psi^{*N} \in L^1(dE)$. Notably,

$$\mathcal H_\mathrm{ac} \subset \mathcal H_\mathrm{eac},$$

and models such as the Fibonacci quasicrystal can satisfy $\mu_\psi^{*N} \ll dE$ even when $\mu_\psi$ is purely singular. This extension is critical for delocalization arguments based on phase-averaged bounds.

3. Sharp Phase-Averaged Delocalization Theorems

The spectral density of the Fibonacci Hamiltonian admits a uniform power-law decay [Le25]:

$|\hat\mu(t)| \leq C |t|^{-\epsilon},\quad|t|\gg1,$

where $\epsilon = \epsilon(V) > 0$ for $0 < V < V_0$. Setting $N := \lceil 1/\epsilon \rceil$ ensures $\mu^{*N} \in L^2(dE) \subset L^1(dE)$, producing the following bounds:

• Lower: $M_p(T) \ge C_1(p, V)\, T^{p(1-1/N)}$
• Upper: $M_p(T) \le C_2(p, V)\, T^{p}$

Consequently,

$\beta^-(p) \ge 1 - 1/N, \qquad \beta^+(p)=1,$

implying almost ballistic quantum transport ($\beta^-(p) \to 1$ as $N \to \infty$) for all $p > 0$ under the mere eventual absolute continuity condition (Leclerc, 7 Dec 2025). This substantially exceeds earlier bounds, which only established $\beta^-(p)>0$ for all $p$.

4. Proof Strategies: Tensor-Product RAGE and Spectral Calculus

The argument leverages Fourier decay and high-order tensor products:

• Step 1: Given $|\hat\mu(t)|\le C|t|^{-\epsilon}$, $\mu^{*N}\in L^2$ follows by repeated convolution.
• Step 2: Construction of an $N$-particle Hamiltonian $H_{\vec\omega}$ yields spectral measure $\mu_{\omega_1}*\cdots*\mu_{\omega_N}$, which after phase-averaging on $\mathbb T^N$ ensures decay in the averaged quantum amplitude by the RAGE theorem and Riemann-Lebesgue lemma.
• Step 3: Cauchy-Schwarz and Hölder inequalities relate phase-averaged moments to averaged amplitudes over higher dimensions, bounding the single-particle behavior.
• Step 4: Sites $|n|\sim T^{1-1/N}$ sustain nontrivial quantum probability, with $⟨|a(n,T;\omega)|^2⟩_\omega \lesssim T^{-2/N}$, enforcing spatial spreading at least to the scale $T^{1-1/N}$.

5. Explicit Asymptotics and Model Dependence

The Fourier decay exponent $\epsilon(V)$ can be chosen proportional to a positive power of $V$ for small $V$:

$$N = \left\lceil 1/\epsilon(V) \right\rceil = O(1/V^\alpha), \quad \text{for some } \alpha>0,$$

yielding

$\beta^-(p) \gtrsim 1 - \epsilon(V)$

and ballisticity $\beta^-(p)\to1$ as $V\to 0$. In the small-coupling regime, the phase-averaged bounds classify quantum transport as essentially ballistic, vastly refining prior delocalization exponents (Leclerc, 7 Dec 2025).

6. Extension to Other Models and Fluctuation-Averaging

A related fluctuation-averaging methodology is employed in the delocalization analysis of random band matrices (Yang et al., 2018). There, high-probability bounds on averages of polynomials in the resolvent entries for random band matrices of size $N$ and bandwidth $W$ enable sharp control of eigenvector delocalization:

Model	Averaging Approach	Principal Bound
Fibonacci Hamiltonian	Phase (Lebesgue average)	$M_p(T) \ge C_1(p, V)\, T^{p(1-1/N)}$
Random Band Matrix	Fluctuation-Averaging	$|G_{xy} - m\delta_{xy}|\prec W^{-d/2}$ at mesoscopic $\eta$

Random band matrices in dimensions $d \geq 2$ realize complete averaged delocalization in the bulk for $N \prec W^{1+d/2}$, ensuring a vanishing fraction of eigenvectors localized at scales $l \gg W$.

7. Research Context and Implications

The combination of explicit spectral Fourier decay, phase-averaged techniques, and the novel eventual absolute continuity condition delineates a powerful regime for establishing ballistic transport in quasicrystal models. For random band matrices, fluctuation-averaging similarly yields universality and delocalization in the bulk spectrum. These frameworks unify phase-averaged quantum dynamics and averaged resolvent methodology, bridging deterministic and random matrix approaches to delocalization phenomena.

This suggests that phase-averaged delocalization bounds, driven by high-order spectral convolution and tensor-product techniques, provide robust quantitative transport measures across both quasiperiodic and random Hamiltonian models.