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Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks

Xinyu Yang, Long Li, Qi Zhou

Abstract

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.

Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks

Abstract

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.
Paper Structure (20 sections, 35 theorems, 180 equations, 1 figure)

This paper contains 20 sections, 35 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantum Walks and Quantum Search Algorithm
  3. Model and result
  4. Preliminaries
  5. Quasiperiodic Cocycle
  6. Avila's global theory
  7. Szegő Cocycle
  8. Aubry-André duality
  9. Lyapunov exponents and parameter regions of the cocycle
  10. Szegő cocycles maps
  11. Nonsingular case
  12. Singular case
  13. Absolutely continuous spectrum in the subcritical region
  14. Singular continuous spectrum
  15. Singular continuous spectrum in regime ii@
  16. ...and 5 more sections

Key Result

Theorem 1.1

Consider the quantum walk $W_{\lambda_1,\lambda_2,\theta,\Phi}$ associated with the conditional shift $S_{\lambda_1}$ and dynamically defined coins eq.coinOp with sampling functions eq.FuncF and eq.FuncG. Assume $t\neq 0$. Then the following hold: If either $\lambda_{1}$ or $\lambda_{2}$ lies in $\left(0,\frac{|1-t^{2}|}{1+t^{2}}\right)$ and $\Phi\in DC$, then:

Figures (1)

  • Figure 1: In area i@, the spectral type is purely absolutely continuous. In area ii@ and on the blue lines, the spectral type is purely singular continuous. In area iii@, the spectral type is purely point.

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1: JM12, BourgainJito
  • Theorem 2.2: Avila2015Acta
  • Theorem 2.3: Avila2015Acta
  • Lemma 2.4: CFLOZ24
  • Remark 2.5
  • Lemma 3.1
  • ...and 57 more