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Exact new mobility edges

Yongjian Wang, Qi Zhou

TL;DR

This work establishes two exact new mobility edges in physically realistic quasiperiodic models by analyzing singular Jacobi operators on the strip. It presents Type II ME (critical versus localized) for a mosaic-type singular Jacobi operator and Type III ME (critical versus extended) for a tailored long-range operator, using a unifying strip-operator framework and Aubry duality. key tools include explicit Lyapunov exponent formulas, Thouless theory, the IDS–rotation-number relation, Jitomirskaya–Last inequalities, and nonperturbative localization techniques. The results illuminate how singularity and duality drive coexistence and transitions among localized, critical, and extended spectral types, with potential experimental relevance for mosaic-like and related systems.

Abstract

Mobility edges (ME), defined as critical energies that separate the extended states from the localized states, are a significant topic in quantum physics. In this paper, we demonstrate the existence of two exact new mobility edges for two physically realistic models: the first, referred to as Type II ME, represents the critical energy that separates the critical states from localized states; the second, referred to as Type III ME, marks the critical energy that separate the critical states from extended states. The proof is based on spectral analysis of singular Jacobi operator on the strip.

Exact new mobility edges

TL;DR

This work establishes two exact new mobility edges in physically realistic quasiperiodic models by analyzing singular Jacobi operators on the strip. It presents Type II ME (critical versus localized) for a mosaic-type singular Jacobi operator and Type III ME (critical versus extended) for a tailored long-range operator, using a unifying strip-operator framework and Aubry duality. key tools include explicit Lyapunov exponent formulas, Thouless theory, the IDS–rotation-number relation, Jitomirskaya–Last inequalities, and nonperturbative localization techniques. The results illuminate how singularity and duality drive coexistence and transitions among localized, critical, and extended spectral types, with potential experimental relevance for mosaic-like and related systems.

Abstract

Mobility edges (ME), defined as critical energies that separate the extended states from the localized states, are a significant topic in quantum physics. In this paper, we demonstrate the existence of two exact new mobility edges for two physically realistic models: the first, referred to as Type II ME, represents the critical energy that separates the critical states from localized states; the second, referred to as Type III ME, marks the critical energy that separate the critical states from extended states. The proof is based on spectral analysis of singular Jacobi operator on the strip.

Paper Structure

This paper contains 40 sections, 51 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Type I ME
  4. Type II ME
  5. Type III ME
  6. Novelty
  7. Singular Jacobi operator on the strip.
  8. Aubry duality.
  9. Difficulty, ideas of the proof
  10. Singular Continuous spectrum.
  11. Absolutely continuous spectrum.
  12. Anderson localization.
  13. Preliminaries
  14. Continued Fraction Expansion.
  15. Cocycle, Lyapunov exponent
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\alpha\in DC$ and $\lambda>0$. For almost every $\theta\in{\mathbb T}$, we have the following:

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2: Quantization of accelerationA4jitomirskaya2012analyticjit2013
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 83 more