[Paper Review] Dynamical Localization for the Random Dimer Model

Stephan De Bièvre, François Germinet|arXiv (Cornell University)|Jul 7, 1999

Spectral Theory in Mathematical Physics18 references45 citations

TL;DR

This paper establishes dynamical localization for the one-dimensional random dimer model with Bernoulli-distributed dimer potentials. Using Lyapunov exponent analysis and multiscale techniques, it proves that eigenfunctions are semi-uniformly exponentially localized away from critical energies, leading to uniform bounds on the time-averaged moments of the position operator, thus confirming dynamical localization for all V > 1 and V ≠ √2, and for V ≤ 1 away from ±V and additional critical points at V = 1/√2 and V = √2.

ABSTRACT

We study the one-dimensional random dimer model, with Hamiltonian $H_\omega=\Delta + V_\omega$, where for all $x\in\Z, V_\omega(2x)=V_\omega(2x+1)$ and where the $V_\omega(2x)$ are i.i.d. Bernoulli random variables taking the values $\pm V, V>0$. We show that, for all values of $V$ and with probability one in $\omega$, the spectrum of $H$ is pure point. If $V\leq1$ and $V eq 1/\sqrt{2}$, the Lyapounov exponent vanishes only at the two critical energies given by $E=\pm V$. For the particular value $V=1/\sqrt{2}$, respectively $V=\sqrt{2}$, we show the existence of additional critical energies at $E=\pm 3/\sqrt{2}$, resp. E=0. On any compact interval $I$ not containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all $q>0$ and for all $\psi\in\ell^2(\Z)$ with sufficiently rapid decrease: $$ \sup_t r^{(q)}_{\psi,I}(t) \equiv \sup_t < P_I(H_\omega)\psi_t, |X|^q P_I(H_\omega)\psi_t > <\infty. $$ Here $\psi_t=e^{-iH_\omega t} \psi$, and $P_I(H_\omega)$ is the spectral projector of $H_\omega$ onto the interval $I$. In particular if $V>1$ and $V eq \sqrt{2}$, these results hold on the entire spectrum (so that one can take $I=\sigma(H_\omega)$).

Motivation & Objective

• To rigorously establish dynamical localization in the one-dimensional random dimer model with i.i.d. Bernoulli-distributed dimer potentials.
• To analyze the Lyapunov exponent and identify critical energies where it vanishes, particularly for special values V = 1/√2 and V = √2.
• To show that eigenfunctions are semi-uniformly exponentially localized away from critical energies, implying dynamical localization.
• To clarify the conditions under which superdiffusive behavior (t^{3/2}) might occur, by ruling out delocalization in the infinite system.
• To extend the understanding of localization-delocalization transitions in random dimer models beyond the standard ±V critical points.

Proposed method

• Analyzes the two-step transfer matrix T^E_v = (S^E_v)^2 for the dimer potential, where S^E_v is the standard one-step transfer matrix.
• Applies the Furstenberg and Kesten theorem to establish the existence and almost-sure constancy of the Lyapunov exponent γ(E).
• Uses the multiscale analysis framework via Wegner-type estimates, relying on the strict positivity of γ(E) away from critical energies.
• Introduces a sequence of random variables Uk and Vk to model the growth of matrix products, tracking the number of Tα and Tβ matrices in the chain.
• Employs the Cauchy-Schwarz inequality and moment estimates on Vm(n) to bound the expected norm growth and show E(|Vm(n)|/n) → 0, implying γ = 0 at critical energies.
• For special cases V = 1/√2 and V = √2, constructs a reduced product involving TαTβ and shows that the norm growth is sub-exponential, leading to vanishing Lyapunov exponent at new critical energies.

Experimental results

Research questions

• RQ1 Under what conditions is the random dimer model dynamically localized, i.e., with uniformly bounded time-averaged moments of the position operator?
• RQ2 Where do the Lyapunov exponent vanish, and what are the implications for eigenfunction localization and spectral type?
• RQ3 Are there additional critical energies beyond E = ±V where the Lyapunov exponent vanishes, and how do they differ from the standard ±V case?
• RQ4 Can the observed t^{3/2} superdiffusive behavior in finite systems be reconciled with dynamical localization in the infinite system?
• RQ5 How does the structure of the dimer potential affect the growth of transfer matrix products and the resulting localization properties?

Key findings

• For all V > 0 and almost all ω, the spectrum of the random dimer Hamiltonian is pure point.
• When V ≤ 1 and V ≠ 1/√2, the Lyapunov exponent vanishes only at E = ±V; for V = 1/√2, additional critical energies appear at E = ±3/√2.
• When V = √2, the Lyapunov exponent vanishes at E = 0, indicating a new critical energy not present in the standard model.
• Away from critical energies, eigenfunctions are semi-uniformly exponentially localized: |ϕ_n,ω(x)| ≤ C(ω,ε,γ) e^{|x_n,ω|^ε} e^{-γ|x - x_n,ω|} for any γ < γ(I).
• This localization implies dynamical localization: for all q > 0 and exponentially decaying initial states ψ, sup_t ⟨PI(Hω)ψ_t, |X|^q PI(Hω)ψ_t⟩ < ∞ almost surely.
• For V > 1 and V ≠ √2, the results hold on the entire spectrum, so the system is dynamically localized globally.