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Convergence of the Planewave Approximations for Quantum Incommensurate Systems

Ting Wang, Huajie Chen, Aihui Zhou, Yuzhi Zhou, Daniel Massatt

TL;DR

The paper develops a rigorous framework for the density of states (DoS) and local density of states (LDoS) in incommensurate (quasi-periodic) Schrödinger systems formed by stacking misoriented 2D layers. It proves the thermodynamic limit of DoS in a weak sense and introduces two efficient planewave-based discretizations: (i) LDoS in reciprocal space with a two-direction cutoff and trapezoidal quadrature, and (ii) a direct planewave discretization without shifting. The authors provide explicit exponential convergence bounds for the first scheme and a convergent, rate-controlled bound for the second, supported by numerical experiments in 1D and 2D that confirm the theory. The methodology enables accurate, scalable computation of electronic observables for incommensurate materials and lays a foundation for extending to nonlinear frameworks such as density functional theory (DFT).

Abstract

Incommensurate structures arise from stacking single layers of low-dimensional materials on top of one another with misalignment such as an in-plane twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. In this paper, we characterize the density of states of Schrödinger operators in the weak sense for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.

Convergence of the Planewave Approximations for Quantum Incommensurate Systems

TL;DR

The paper develops a rigorous framework for the density of states (DoS) and local density of states (LDoS) in incommensurate (quasi-periodic) Schrödinger systems formed by stacking misoriented 2D layers. It proves the thermodynamic limit of DoS in a weak sense and introduces two efficient planewave-based discretizations: (i) LDoS in reciprocal space with a two-direction cutoff and trapezoidal quadrature, and (ii) a direct planewave discretization without shifting. The authors provide explicit exponential convergence bounds for the first scheme and a convergent, rate-controlled bound for the second, supported by numerical experiments in 1D and 2D that confirm the theory. The methodology enables accurate, scalable computation of electronic observables for incommensurate materials and lays a foundation for extending to nonlinear frameworks such as density functional theory (DFT).

Abstract

Incommensurate structures arise from stacking single layers of low-dimensional materials on top of one another with misalignment such as an in-plane twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. In this paper, we characterize the density of states of Schrödinger operators in the weak sense for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.
Paper Structure (14 sections, 9 theorems, 126 equations, 7 figures)

This paper contains 14 sections, 9 theorems, 126 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Schrödinger operator for incommensurate systems
  3. Thermodynamic limit of the density of states
  4. Planewave approximations and convergence analysis
  5. Planewave discretization based on LDoS in reciprocal space
  6. Planewave discretization of the Hamiltonian without shifting
  7. Numerical experiments
  8. Conclusions
  9. Justification and decay property of $g(\hat{H}({\xi}))$
  10. Proofs
  11. Proof of Lemma \ref{['lem:boundtracejk']}
  12. Proof of Theorem \ref{['theo:tdllimit']}
  13. Proof of Theorem \ref{['theo:convergence_discretizedDos']}
  14. Proof of Theorem \ref{['theorem:WLconvergence-dos']}

Key Result

Lemma 3.1

\newlabellem:boundtracejk0 Let $j,k\in \mathbb{Z}^d$ and $g \in \Lambda_{\zeta,\delta}$. Then the operator $\chi_j g(H)\chi_k$ is trace class, and it has a continuous kernel $K_{jk}(x,y)$ with Moreover, there exists a constant $C>0$ independent of $j$ and $k$ such that

Figures (7)

  • Figure 1: The unit cells $\Gamma_j$ are shaded and their corresponding lattices $\mathcal{R}_j$ are displayed. Configuration vectors $b_1$ and $b_2$ corresponding to $x \in \mathbb{R}^2$ (central dot in both unit cells).
  • Figure 1: Schematic plot of the domain $\mathcal{D}_{W,L}$ in the reciprocal space.
  • Figure 1: (Example 1) Error decay with respect to the planewave cutoffs. Left: convergence w.r.t. $L$ by \ref{['discreteddos']}. Middle: convergence w.r.t. $W$ by \ref{['discreteddos']}. Right: convergence by \ref{['def:dos-reciprocalWL']}.
  • Figure 1: For proof of Lemma \ref{['lemma:tdl:gH00']}, we consider $\xi_{\bf G} = \xi + G_1 + G_2$ and $\xi_{G'} = \xi + G_1' +G_2'$ as displayed. Left: the case of $|\xi_{\bf G} - \xi_{\bf G'}| < \bar{R}/2$. Right: the case of $|\xi_{\bf G} - \xi_{\bf G'}| \geq \bar{R}/2$.
  • Figure 2: (Example 1) Error decay with respect to the quadrature mesh size $h$ by scheme \ref{['discreteddos']}. Left: $W=25.0$. Right: $W=5.0$.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.2: connection between $\hat{H}(\xi)$ and $H$
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4: convergence rate of $\pmb{\phi}(L)$
  • Remark 4.5: intuition behind the planewave cutoffs
  • Remark 4.6: comparison between the two algorithms
  • ...and 12 more