---
title: Phase-Averaged Delocalization Bounds
url: https://www.emergentmind.com/topics/phase-averaged-delocalization-bounds
type: topic
---

# Phase-Averaged Delocalization Bounds

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 [2512.06844, 1807.02447].

## 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 [2512.06844] 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 [2512.06844]. 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 [2512.06844].

## 6. Extension to Other Models and Fluctuation-Averaging

A related fluctuation-averaging methodology is employed in the delocalization analysis of random band matrices [1807.02447]. 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.*

Source: https://www.emergentmind.com/topics/phase-averaged-delocalization-bounds