Chebyshev Accelerated Subspsace Eigensolver for Pseudo-hermitian Hamiltonians

TL;DR

The paper addresses the computational challenge of extracting thousands of eigenpairs from large pseudo-hermitian Hamiltonians that arise in Bethe-Salpeter treatments of excitonic materials. It extends ChASE by developing an oblique Rayleigh–Ritz framework with a dual subspace to preserve biorthogonality, and by constructing a Hermitian-spectral-equivalent reduction that maintains quadratic convergence and robust performance on massively parallel GPUs. The authors provide a comprehensive numerical analysis and large-scale GPU experiments showing that the pseudo-hermitian extension achieves convergence rates and parallel efficiency comparable to the Hermitian case, while handling the additional spectral structure inherent to $H$ and $SH$. This work enables scalable, accurate solution of thousands of extremal eigenpairs in excitonic simulations, with practical impact for materials science and optoelectronic modeling.

Abstract

Studying the optoelectronic structure of materials can require the computation of up to several thousands of the smallest eigenpairs of a pseudo-hermitian Hamiltonian. Iterative eigensolvers may be preferred over direct methods for this task since their complexity is a function of the desired fraction of the spectrum. In addition, they generally rely on highly optimized and scalable kernels such as matrix-vector multiplications that leverage the massive parallelism and the computational power of modern exascale systems. \textit{Chebyshev Accelerated Subspace iteration Eigensolver} (ChASE) is able to compute several thousands of the most extreme eigenpairs of dense hermitian matrices with proven scalability over massive parallel accelerated clusters. This work presents an extension of ChASE to solve for a portion of the spectrum of pseudo-hermitian Hamiltonians as they appear in the treatment of excitonic materials. The new pseudo-hermitian solver achieves similar convergence and performance as the hermitian one. By exploiting the numerical structure and spectral properties of the Hamiltonian matrix, we propose an oblique variant of Rayleigh-Ritz projection featuring quadratic convergence of the Ritz-values with no explicit construction of the dual basis set. Additionally, we introduce a parallel implementation of the recursive matrix-product operation appearing in the Chebyshev filter with limited amount of global communications. Our development is supported by a full numerical analysis and experimental tests.

Figures (8)

• Figure 1: Illustration of the effect of the Chebsyhev polynomial filter - the gray filling corresponds to the filtered area and the dashed pattern represents the area of target eigenpairs.
• Figure 1: Eigenvalues of $H$ when $SH$ is indefinite vs. HPD - The brackets and the dashed square represent the bounds of the fields of values ${\mathcal{F}}(H)$
• Figure 1: First $100^{\text{th}}$
• Figure 1: ${\texttt{tol}}=10^{-8}$
• Figure 2: Last $100^{\text{th}}$

Theorems & Definitions (29)

• Theorem 3.1
• Theorem 3.2
• Proof 1
• Theorem 3.3
• Proof 2
• Lemma 3.4
• Theorem 3.5
• Proof 3
• Theorem 5.1
• Proof 4