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Arithmetic spectral transition for the unitary almost Mathieu operator

Fan Yang

TL;DR

This work analyzes the unitary almost Mathieu operator (UAMO) in the positive Lyapunov exponent regime, establishing a sharp arithmetic localization threshold: Anderson localization holds for irrational frequencies with $\beta(\omega)<L$ and non-resonant phases $\tilde{\gamma}(\omega,\theta)=0$. By embedding the model into extended CMV matrices and exploiting Szegő cocycles, the authors perform a scale-by-scale Green's function analysis along continued-fraction scales $q_n$, balancing Lyapunov growth against small divisors. They distinguish strong and weak Liouville scales and treat non-resonant and resonant regimes accordingly, proving exponential decay of generalized eigenfunctions and hence pure point spectrum with exponentially decaying eigenfunctions. The results extend prior Diophantine-frequency localization to a sharp arithmetic threshold, with potential implications for sharp spectral transitions in unitary quasi-periodic systems and quantum walks in magnetic fields.

Abstract

We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le λ_1<λ_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $β(ω)$. More precisely, for every irrational $ω$ with $β(ω)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $θ$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $β(ω)=0$) to a sharp threshold in frequency.

Arithmetic spectral transition for the unitary almost Mathieu operator

TL;DR

This work analyzes the unitary almost Mathieu operator (UAMO) in the positive Lyapunov exponent regime, establishing a sharp arithmetic localization threshold: Anderson localization holds for irrational frequencies with and non-resonant phases . By embedding the model into extended CMV matrices and exploiting Szegő cocycles, the authors perform a scale-by-scale Green's function analysis along continued-fraction scales , balancing Lyapunov growth against small divisors. They distinguish strong and weak Liouville scales and treat non-resonant and resonant regimes accordingly, proving exponential decay of generalized eigenfunctions and hence pure point spectrum with exponentially decaying eigenfunctions. The results extend prior Diophantine-frequency localization to a sharp arithmetic threshold, with potential implications for sharp spectral transitions in unitary quasi-periodic systems and quantum walks in magnetic fields.

Abstract

We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on in a homogeneous magnetic field. In the positive Lyapunov exponent regime , we establish an arithmetic localization statement governed by the frequency exponent . More precisely, for every irrational with , where denotes the Lyapunov exponent, and every non-resonant phase , we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which ) to a sharp threshold in frequency.
Paper Structure (26 sections, 32 theorems, 187 equations)

This paper contains 26 sections, 32 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The unitary almost Mathieu operator
  3. Measure-theoretic vs. arithmetic localization
  4. Arithmetic input and main result
  5. Strategy and organization
  6. Preliminaries
  7. Finite-volume restrictions
  8. Extended CMV background
  9. Szegő cocycle and Lyapunov exponent
  10. Continued fractions and arithmetic exponents
  11. Poisson formula and Szegő connection
  12. Preliminary lemmas
  13. An outline of the proof
  14. Eigenfunction in the non-resonant regimes in the strong Liouville case
  15. Estimates of $\sum_2$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assume eq:supercritical_regime_intro. Let $\omega\in\mathbb R\setminus\mathbb Q$ satisfy $\beta(\omega)<L$, and let $\theta\in\mathbb T$ be non-resonant, i.e. $\tilde{\gamma}(\omega,\theta)=0$. Then the unitary almost Mathieu operator $W_{\lambda_1,\lambda_2,\omega,\theta}$ has Anderson localization

Theorems & Definitions (53)

  • Theorem 1.1: Arithmetic Anderson localization
  • Remark 1.2
  • Theorem 2.1: Corollary 2.8 of CFO and (3.25) of FanUAMO
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 3.1: Lagrange interpolation
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 5.1
  • ...and 43 more