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Learning the local density of states of a bilayer moiré material in one dimension

Diyi Liu, Alexander B. Watson, Michael Hott, Stephen Carr, Mitchell Luskin

TL;DR

The paper develops a rigorous framework for learning the local density of states of a one-dimensional bilayer moiré material under twist. It formalizes the twist operator as the composition of an inverse map from LDOS images to tight-binding parameters with the forward LDOS map at nonzero twist, and proves existence and continuity under precise conditions. By invoking the universal approximation theorem, it shows that the twist operator can be effectively approximated by a two-layer neural network, justifying data-driven operator learning for moiré systems. The work also provides detailed numerical analysis and error bounds for computing LDOS images in commensurate setups, ensuring that machine-learning approaches are backed by solid convergence results. Altogether, it establishes a principled path from untwisted electronic structure data to twisted properties, with potential extensions to higher-dimensional and more complex heterostructures.

Abstract

Recent work of three of the authors showed that the operator which maps the local density of states of a one-dimensional untwisted bilayer material to the local density of states of the same bilayer material at non-zero twist, known as the twist operator, can be learned by a neural network. In this work, we first provide a mathematical formulation of that work, making the relevant models and operator learning problem precise. We then prove that the operator learning problem is well-posed for a family of one-dimensional models. To do this, we first prove existence and regularity of the twist operator by solving an inverse problem. We then invoke the universal approximation theorem for operators to prove existence of a neural network capable of approximating the twist operator.

Learning the local density of states of a bilayer moiré material in one dimension

TL;DR

The paper develops a rigorous framework for learning the local density of states of a one-dimensional bilayer moiré material under twist. It formalizes the twist operator as the composition of an inverse map from LDOS images to tight-binding parameters with the forward LDOS map at nonzero twist, and proves existence and continuity under precise conditions. By invoking the universal approximation theorem, it shows that the twist operator can be effectively approximated by a two-layer neural network, justifying data-driven operator learning for moiré systems. The work also provides detailed numerical analysis and error bounds for computing LDOS images in commensurate setups, ensuring that machine-learning approaches are backed by solid convergence results. Altogether, it establishes a principled path from untwisted electronic structure data to twisted properties, with potential extensions to higher-dimensional and more complex heterostructures.

Abstract

Recent work of three of the authors showed that the operator which maps the local density of states of a one-dimensional untwisted bilayer material to the local density of states of the same bilayer material at non-zero twist, known as the twist operator, can be learned by a neural network. In this work, we first provide a mathematical formulation of that work, making the relevant models and operator learning problem precise. We then prove that the operator learning problem is well-posed for a family of one-dimensional models. To do this, we first prove existence and regularity of the twist operator by solving an inverse problem. We then invoke the universal approximation theorem for operators to prove existence of a neural network capable of approximating the twist operator.
Paper Structure (27 sections, 16 theorems, 137 equations, 4 figures)

This paper contains 27 sections, 16 theorems, 137 equations, 4 figures.

Table of Contents

  1. Motivation and summary
  2. Related work
  3. Structure of this work
  4. Mathematical formulation of the coupled chain model
  5. The tight binding model
  6. Stacking-dependent local density of states: analysis and computation
  7. Stacking-dependent local density of states
  8. The local density of states image
  9. Introduction to the twist operator
  10. Existence of the twist operator: an inverse problem perspective
  11. Rigorous theory of the twist operator: precise inverse problem
  12. Rigorous theory of the twist operator: solving the inverse problem
  13. Universal approximation with neural network
  14. Numerical details of computing LDOS images
  15. The commensurate coupled chain
  16. ...and 12 more sections

Key Result

Theorem 4.2

Existence and Stability of Inverse Map \newlabelthm:in_specific0 Assume $N_E > n_{\text{poly}}$, where $n_{\text{poly}}$ is the degree of the polynomial $g$ in equ:ldos. Let $E_{N_E} > E_{1}$ be otherwise arbitrary. Let $N_d \geq 3$, and assume that the points $0,\frac{1}{4},\frac{1}{2}$ are includ Assume further that the interlayer coupling function is given by eq:inter, with $f$ exponential, so

Figures (4)

  • Figure 1: Schematic of the coupled chain system described in Section \ref{['sec:models']} with two orbitals per cell ($m_1 = m_2 = 2$), with a positive lattice mismatch $\theta > 0$ without interlayer shift ($d = 0$). When $\theta$ is irrational, the system has no exact periodic cell.
  • Figure 1: Schematic showing relationships between tight-binding models of untwisted and twisted structures and stacking-dependent LDOS images. The twist operator $\mathcal{L}_\theta$ learned in Liu_2022 maps from untwisted LDOS images to twisted LDOS images. It can be defined via the inverse map from untwisted LDOS images to tight-binding model parameters, composed with the forward map from these tight-binding model parameters to the LDOS image of the twisted bilayer tight-binding model.
  • Figure 1: Schematic of the commensurate coupled chain system described in Section \ref{['sec:commensurate']} with two orbitals per cell ($m_1 = 1, m_2 = 2$), with a rational lattice mismatch $\theta = \frac{1}{3}$ without interlayer shift ($d = 0$). Despite the lattice mismatch, the system is periodic with supercell period $2$.
  • Figure 2: Schematic of the coupled chain system described in Section \ref{['sec:models']} with two orbitals per cell ($m_1 = m_2 = 2$), with a positive lattice mismatch $\theta > 0$ with interlayer shift ($d \neq 0$).

Theorems & Definitions (42)

  • Remark 4.1
  • Theorem 4.2
  • Proof 1
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • Proof 2
  • Remark 4.6
  • Remark 4.7
  • Remark 4.8
  • ...and 32 more