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A high-order regularized delta-Chebyshev method for computing spectral densities

Jinjing Yi, Daniel Massatt, Andrew Horning, Mitchell Luskin, J. H. Pixley, Jason Kaye

Abstract

We introduce a numerical method for computing spectral densities, and apply it to the evaluation of the local density of states (LDOS) of sparse Hamiltonians derived from tight-binding models. The approach, which we call the high-order delta-Chebyshev method, can be viewed as a variant of the popular regularized Chebyshev kernel polynomial method (KPM), but it uses a high-order accurate approximation of the $δ$-function to achieve rapid convergence to the thermodynamic limit for smooth spectral densities. The costly computational steps are identical to those for KPM, with high-order accuracy achieved by an inexpensive post-processing procedure. We apply the algorithm to tight-binding models of graphene and twisted bilayer graphene, demonstrating high-order convergence to the LDOS at non-singular points.

A high-order regularized delta-Chebyshev method for computing spectral densities

Abstract

We introduce a numerical method for computing spectral densities, and apply it to the evaluation of the local density of states (LDOS) of sparse Hamiltonians derived from tight-binding models. The approach, which we call the high-order delta-Chebyshev method, can be viewed as a variant of the popular regularized Chebyshev kernel polynomial method (KPM), but it uses a high-order accurate approximation of the -function to achieve rapid convergence to the thermodynamic limit for smooth spectral densities. The costly computational steps are identical to those for KPM, with high-order accuracy achieved by an inexpensive post-processing procedure. We apply the algorithm to tight-binding models of graphene and twisted bilayer graphene, demonstrating high-order convergence to the LDOS at non-singular points.

Paper Structure

This paper contains 9 sections, 1 theorem, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Kernel polynomial method
  3. High-order regularized Delta-Chebyshev method
  4. Numerical examples
  5. Graphene
  6. Twisted Bilayer Graphene
  7. Conclusion
  8. Convergence of the kernel polynomial method with Jackson regularization
  9. High-order kernels

Key Result

Theorem 1

Let $\mu$ be a probability measure on the real line that is absolutely continuous on an interval $I=(E-\delta,E+\delta)$ for some fixed $E$ and $\delta>0$. Denote the Radon--Nikodym derivative of the absolutely continuous component of $\mu$ by $\rho$, and suppose that $\rho$ is $n$-times continuousl

Figures (8)

  • Figure 1: Monolayer graphene lattice with primitive lattice vectors $v_i$ and supercell lattice vectors $Lv_i$. We illustrate for the highlighted site the corresponding $R_N$, the distance from the site to the edge of the domain.
  • Figure 2: Dirichlet and Jackson kernels $k_p(E, 0)$ with $p = 50$, and high-order kernel $K_\eta(E,0)$ with $m = 6$ and $\eta = 0.05$.
  • Figure 3: Density of states for nearest-neighbor tight-binding model of graphene produced by (a) Jackson KPM and (b) the HODC method, for several values of (a) the Chebyshev degree parameter $p$ and (b) the regularization parameter $\eta$. Dashed black line shows the exact density of states.
  • Figure 4: Error of the HODC scheme for the graphene example at $E = 0.5$. (Top) $\mathcal{O}(\eta^m)$ convergence for $m = 6$, using fixed, sufficiently large Chebyshev degree $p$. For sufficiently small $\eta$, finite size effects amplify the error, requiring larger system sizes $L$. (Bottom) Convergence of Jackson KPM and HODC with $m = 6$. For HODC, $\eta$ is varied, and the Chebyshev order $p$ is adaptively selected based on an error tolerance $\epsilon = 10^{-12}$.
  • Figure 5: A cut-out of TBG of radius $R_N$ centered at the site ${\bf r}$. The shaded parallelogram corresponds to a moiré unit cell.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1