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Ergodic properties of one-dimensional incommensurate bilayer materials

Nathan J. Essner, Jeremiah Williams, Alexander B. Watson

Abstract

We consider one-dimensional deterministic and random tight-binding Hamiltonians modeling electronic properties of twisted bilayer materials. When the twisted structure is incommensurate, we prove convergence of the density of states measure in the thermodynamic limit and Pastur's theorem on shift-invariance of the spectrum. Our results extend those of Massatt et al. and Cancès et al. in allowing for randomness. We provide numerical density of states computations for the operators we consider.

Ergodic properties of one-dimensional incommensurate bilayer materials

Abstract

We consider one-dimensional deterministic and random tight-binding Hamiltonians modeling electronic properties of twisted bilayer materials. When the twisted structure is incommensurate, we prove convergence of the density of states measure in the thermodynamic limit and Pastur's theorem on shift-invariance of the spectrum. Our results extend those of Massatt et al. and Cancès et al. in allowing for randomness. We provide numerical density of states computations for the operators we consider.
Paper Structure (22 sections, 11 theorems, 109 equations, 7 figures)

This paper contains 22 sections, 11 theorems, 109 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Structure of work
  5. Related work
  6. Incommensurate bilayer materials: 1D tight-binding models
  7. Model of incommensurate coupled chains
  8. Reduced model of incommensurate coupled chains
  9. Connection to almost Mathieu operator
  10. Model of incommensurate coupled chains with random onsite potentials
  11. Incommensurate bilayer materials: ergodic properties
  12. Definition of an ergodic operator
  13. Reduced incommensurate coupled chain model as an ergodic operator
  14. Convergence of the density of states measure for the incommensurate coupled chain model
  15. Almost-sure spectrum for the incommensurate coupled chain model
  16. ...and 7 more sections

Key Result

Theorem 3.5

Consider the graph $\mathbb{Z}$ endowed with the group of graph automorphisms $\mathcal{S} = (S_x)_{x \in \mathbb{Z}}$ where $S_x$ translates the graph by $x$, i.e., $S_x n := n - x$. For each $x \in \mathbb{Z}$, let Then, $(T_x)_{x \in \mathbb{Z}}$ is a group of measure-preserving transformations representing $\mathcal{S}$ whose action on the probability space $([0,1 - \theta),\mathcal{A},\mathb

Figures (7)

  • Figure 1: A small $R=10$ demonstration of how the coupled chains are arranged. The center atoms in the chain are aligned at $b=0$ and $b>0$ shifts the entire lower chain.
  • Figure 2: Density of states of the reduced incommensurate coupled chain operator \ref{['eq:reduced_single_chain']} in color against natural log of interatomic distance ratio and energy level.
  • Figure 3: Density of states of the reduced incommensurate coupled chain operator with inverse Laplacian replaced by identity \ref{['eq:reduced_single_chain_noLap']} in color against natural log of interatomic distance ratio and energy level.
  • Figure 4: Higher resolution zoom of Figure \ref{['fig:rcnumerics']} showing chain ratios ranging from 1.25 to 3. Self similarity near the top and bottom of the spectrum suggests a fractal structure, especially for $\ln(p/q) \approx .7 \implies p/q \approx 2$.
  • Figure 5: Density of states of the incommensurate coupled chain operator in color against natural log of interatomic distance ratio and energy level
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Definition 3.6
  • ...and 24 more