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Numerical computation of the density of states of aperiodic multiscale Schrödinger operators

Eric Cancès, Daniel Massatt, Long Meng, Étienne Polack, Xue Quan

TL;DR

This work addresses the DoS computation for incommensurate, multiscale Schrödinger operators by comparing a momentum-space method and a semiclassical ε-expansion on a 1D model. The momentum-space approach uses an unfolded lattice operator Ŝ_ε(ξ) and a Chebyshev/KPM framework to evaluate the DoS via ⟨f(H_ε)⟩ = (2π)^{-d} ∫ [f(Ŝ_ε(ξ))]_{0,0} dξ, while the semiclassical route provides a rigorous ε-expansion Tr(f(H_ε)) = L_0(f) + ε L_1(f) + ε^2 L_2(f) + … derived from Weyl quantization and the Helffer–Sjöstrand formula. Numerical experiments in 1D show strong agreement between the two methods for small ε, but reveal oscillatory deviations at larger ε that are explained by a harmonic-oscillator effective model around moiré-scale Van Hove–like points. The results demonstrate that higher-order semiclassical corrections are important beyond very small ε and offer a principled framework to interpret and predict DoS oscillations in incommensurate systems. Overall, the work provides practical, convergent tools for DoS estimation in moiré materials and deepens the understanding of spectral features linked to two-scale interactions.

Abstract

Computing the electronic structure of incommensurate materials is a central challenge in condensed matter physics, requiring efficient ways to approximate spectral quantities such as the density of states (DoS). In this paper, we numerically investigate two distinct approaches for approximating the DoS of incommensurate Hamiltonians for small values of the incommensurability parameters $ε$ (e.g., small twist angle, or small lattice mismatch): the first employs a momentum-space decomposition, and the second exploits a semiclassical expansion with respect to $ε$. In particular, we compare these two methods using a 1D toy model. We check their consistency by comparing the asymptotic expansion terms of the DoS, and it is shown that, for full DoS, the two methods exhibit good agreement in the small $ε$ limit, while discrepancies arise for less small $ε$, which indicates the importance of higher-order corrections in the semiclassical method for such regimes. We find these discrepancies to be caused by oscillations in the DoS at the semiclassical analogues of Van Hove singularities, which can be explained qualitatively, and quantitatively for $ε$ small enough, by a semiclassical approach.

Numerical computation of the density of states of aperiodic multiscale Schrödinger operators

TL;DR

This work addresses the DoS computation for incommensurate, multiscale Schrödinger operators by comparing a momentum-space method and a semiclassical ε-expansion on a 1D model. The momentum-space approach uses an unfolded lattice operator Ŝ_ε(ξ) and a Chebyshev/KPM framework to evaluate the DoS via ⟨f(H_ε)⟩ = (2π)^{-d} ∫ [f(Ŝ_ε(ξ))]_{0,0} dξ, while the semiclassical route provides a rigorous ε-expansion Tr(f(H_ε)) = L_0(f) + ε L_1(f) + ε^2 L_2(f) + … derived from Weyl quantization and the Helffer–Sjöstrand formula. Numerical experiments in 1D show strong agreement between the two methods for small ε, but reveal oscillatory deviations at larger ε that are explained by a harmonic-oscillator effective model around moiré-scale Van Hove–like points. The results demonstrate that higher-order semiclassical corrections are important beyond very small ε and offer a principled framework to interpret and predict DoS oscillations in incommensurate systems. Overall, the work provides practical, convergent tools for DoS estimation in moiré materials and deepens the understanding of spectral features linked to two-scale interactions.

Abstract

Computing the electronic structure of incommensurate materials is a central challenge in condensed matter physics, requiring efficient ways to approximate spectral quantities such as the density of states (DoS). In this paper, we numerically investigate two distinct approaches for approximating the DoS of incommensurate Hamiltonians for small values of the incommensurability parameters (e.g., small twist angle, or small lattice mismatch): the first employs a momentum-space decomposition, and the second exploits a semiclassical expansion with respect to . In particular, we compare these two methods using a 1D toy model. We check their consistency by comparing the asymptotic expansion terms of the DoS, and it is shown that, for full DoS, the two methods exhibit good agreement in the small limit, while discrepancies arise for less small , which indicates the importance of higher-order corrections in the semiclassical method for such regimes. We find these discrepancies to be caused by oscillations in the DoS at the semiclassical analogues of Van Hove singularities, which can be explained qualitatively, and quantitatively for small enough, by a semiclassical approach.
Paper Structure (21 sections, 4 theorems, 108 equations, 11 figures, 1 table)

This paper contains 21 sections, 4 theorems, 108 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and numerical methods
  3. Momentum-space method
  4. Semiclassical expansion
  5. Density of states of Schrödinger operators in the semiclassical limit
  6. Semiclassical analysis of the two-scale Hamiltonian $H_\epsilon$
  7. Numerical experiments
  8. Discretization
  9. Momentum-space method
  10. Semiclassical expansion
  11. Density of states calculations
  12. Effective model for oscillations in DoS
  13. Conclusion
  14. Derivation of formulae \ref{['eq:L1']} and \ref{['eq:L2']}
  15. Proof of \ref{['eq:L1']}
  16. ...and 6 more sections

Key Result

Proposition 2.1

For $\psi \in \mathcal{S}(\mathbb{R}^n;\mathbb{C})$, $\delta,\zeta > 0$, we have for $f \in \Lambda_{\zeta,\delta}$,

Figures (11)

  • Figure 1: Comparison of two numerical approximations of the function $E \mapsto L_{0,\sigma}(E)$ over the energy range -2020 for $\sigma=0.4$ (Left) and $\sigma=0.08$ (Right). The approximations obtained by the momentum-space method are in solid blue, the ones obtained by discretization of the semiclassical formulae are in dashed orange, and the absolute errors between the two approximations are in black.
  • Figure 2: Comparison of two numerical approximations of the function $E \mapsto L_{1,\sigma}(E)$ over the energy range -2020 for $\sigma=0.4$ (Left) and $\sigma=0.08$ (Right). The approximations obtained by the momentum-space method are in solid blue, the ones obtained by discretization of the semiclassical formulae are in dashed orange, and the absolute errors between the two approximations are in black.
  • Figure 3: Comparison of two numerical approximations of the function $E \mapsto L_{2,\sigma}(E)$ over the energy range -2020 for $\sigma=0.4$ (Left) and $\sigma=0.08$ (Right). The approximations obtained by the momentum-space method are in solid blue, the ones obtained by discretization of the semiclassical formulae are in dashed orange, and the absolute errors between the two approximations are in black.
  • Figure 4: Comparison of the momentum space (solid blue) and second-order semiclassical (dashed orange) approximations of $\nu_{\epsilon,\sigma}$ of $\epsilon=0.001$ over the energy range -2020 for $\sigma=0.4$ (Left) and ${\sigma=0.08}$ (Right).
  • Figure 5: Comparison of the momentum space (solid blue) and second-order semiclassical (dashed orange) approximations of $\nu_{\epsilon,\sigma}$ of $\epsilon=0.01$ over the energy range -2020 for $\sigma=0.4$ (Left) and ${\sigma=0.08}$ (Right).
  • ...and 6 more figures

Theorems & Definitions (6)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6