Efficient Reduced Projection Method for Solving High-Dimensional Quasiperiodic Schrödinger Eigenvalue Problems in Photonic Moiré Lattices

Core Concepts

The authors propose an efficient reduced projection method (RPM) to significantly reduce the degrees of freedom required for solving high-dimensional quasiperiodic Schrödinger eigenvalue problems describing photonic moiré lattices, while maintaining high accuracy.

Abstract

The paper presents a reduced projection method (RPM) for efficiently solving quasiperiodic Schrödinger eigenvalue problems that describe photonic moiré lattices. The key highlights are:

The authors prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. This allows for a significant reduction in the degrees of freedom (DOF) required for the eigenvalue computation.

The RPM is developed by introducing a variational framework and a rigorous error analysis is provided, demonstrating that the RPM can achieve the same level of accuracy as the classical projection method (PM) with much fewer DOF.

Numerical examples in 1D and 2D photonic moiré lattices are presented, showcasing the effectiveness and efficiency of the proposed RPM compared to the PM. The RPM is able to achieve machine precision accuracy with over 80% reduction in DOF.

The computational complexity of the RPM for solving the first k eigenpairs is reduced from O(kN^2n) to O(kN^2(n-d)D^2d), where N is the number of Fourier grids in one direction, n is the dimension of the periodic domain, d is the dimension of the quasiperiodic domain, and D is the truncation parameter in the RPM.

Stats

The 1D quasiperiodic potential is given by v1(z) = E0 / (cos^2(cos(θ/2)z) + cos^2(sin(θ/2)z) + 1), where θ = π/6 and E0 = 1.
The 2D quasiperiodic potential is given by v2(z1, z2) = E0 / (cos^2(cos(θ/2)z1) + cos^2(sin(θ/2)z2) + 1), where θ = π/6 and E0 = 1.

Quotes

"The RPM is able to achieve machine precision accuracy with over 80% reduction in DOF."
"The computational complexity of the RPM for solving the first k eigenpairs is reduced from O(kN^2n) to O(kN^2(n-d)D^2d)."

Key Insights Distilled From

Reduced projection method for photonic moiré lattices

by Zixuan Gao,Z... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2309.09238.pdf

Deeper Inquiries

How can the RPM be extended to solve quasiperiodic problems with more general potentials beyond the moiré lattice structure

The RPM can be extended to solve quasiperiodic problems with more general potentials beyond the moiré lattice structure by adapting the projection matrix and basis space to accommodate the specific characteristics of the new potential functions. In the context of the Schrödinger equation, the RPM relies on the properties of the Fourier coefficients of the eigenfunctions to reduce the degrees of freedom significantly. By considering the decay rate of the generalized Fourier coefficients along a fixed direction associated with the new potential function, the RPM can be tailored to handle a wider range of quasiperiodic potentials. This adaptation may involve adjusting the basis reduction strategy and the error estimates to suit the properties of the new potential function. Additionally, incorporating different projection matrices that capture the essential features of the potential function can enhance the efficiency and accuracy of the RPM in solving diverse quasiperiodic problems.

What are the limitations of the RPM and under what conditions might the classical projection method be more suitable

The limitations of the RPM primarily arise when dealing with highly complex or irregular potential functions that do not exhibit the same decay rate of Fourier coefficients as observed in the moiré lattice structures. In such cases, the RPM may struggle to achieve the same level of accuracy and efficiency, leading to potential errors in the eigenvalue solutions. Additionally, the RPM's effectiveness may be limited in scenarios where the dimensionality of the problem is extremely high, resulting in computational challenges and increased memory consumption. Under these conditions, the classical projection method may be more suitable as it provides a more straightforward approach to solving quasiperiodic problems without the need for extensive basis reduction and specialized error estimates. The classical projection method may offer better performance in scenarios where the properties of the potential function do not align well with the assumptions underlying the RPM.

Can the ideas behind the RPM be applied to solve other types of high-dimensional eigenvalue problems beyond the Schrödinger equation

The ideas behind the RPM can be applied to solve other types of high-dimensional eigenvalue problems beyond the Schrödinger equation by adapting the methodology to suit the specific characteristics of the new problem. The concept of reducing the basis space and exploiting the decay rate of Fourier coefficients can be generalized to various eigenvalue problems in different domains such as quantum mechanics, solid-state physics, and computational chemistry. By identifying the key properties of the eigenfunctions and leveraging efficient reduction strategies, the RPM can be extended to tackle a wide range of high-dimensional eigenvalue problems. The application of the RPM principles to diverse domains may require modifications to the projection matrices, basis spaces, and error estimation techniques to ensure accurate and efficient solutions. Overall, the core principles of the RPM can serve as a foundation for developing innovative approaches to address complex high-dimensional eigenvalue problems in various fields.