High-order Regularized Delta-Chebyshev Method Achieves Rapid Convergence to Spectral Densities for Tight-Binding Models

Computing spectral densities remains a significant challenge in many areas of physics and materials science, and a team led by Jinjing Yi, Daniel Massatt from New Jersey Institute of Technology, and Andrew Horning from Rensselaer Polytechnic Institute now presents a new method to address this problem. Their high-order regularized delta-Chebyshev method offers a powerful alternative to existing techniques, achieving rapid convergence to accurate results for smooth spectral densities while maintaining computational efficiency. The researchers demonstrate the method’s effectiveness by applying it to complex tight-binding models of graphene and twisted bilayer graphene, revealing high-order convergence to the local density of states at points where the spectrum is well-behaved. This advancement, also involving Mitchell Luskin from University of Minnesota, J. H. Pixley, and Jason Kaye from Flatiron Institute, promises to accelerate calculations and deepen understanding of materials with complex electronic structures.

Accurate 2D Material Electronic Structure Calculations

This research focuses on developing highly accurate numerical methods for calculating the electronic properties of two-dimensional materials, including twisted bilayer graphene and related structures. Scientists are pushing the boundaries of computational materials science, striving for reliable predictions of material behaviour and spectral properties like the density of states and transport characteristics. The work also aims to model materials with imperfections and complex arrangements. The research leverages the Kernel Polynomial Method (KPM), providing a powerful way to approximate spectral functions, particularly when dealing with rapidly changing features.

KPM relies on Chebyshev polynomials to efficiently represent and approximate functions, and researchers also utilise Wannier functions, which are maximally localized, to build a simplified, yet accurate, tight-binding model that captures essential physics while reducing computational demands. A sophisticated mathematical framework, involving rigged Hilbert spaces, is employed to rigorously represent and manipulate functions, ensuring the accuracy of calculations involving spectral properties. The team is also developing methods less sensitive to the choice of basis set, leading to more reliable results. Tight-binding models simplify electronic structure calculations while retaining essential physics, and Kubo formulas are used to calculate transport properties like conductivity.

The research has specific applications in understanding twisted bilayer graphene, where the material exhibits unusual electronic properties at certain twist angles. The methods are also applicable to a wider range of 2D materials stacked together to form heterostructures, allowing scientists to investigate transport properties, study how imperfections affect electron localization, and accurately calculate spectral functions. The work also explores the photogalvanic effect, which involves the generation of current in materials due to light. The key contributions of this work lie in the development of mathematically rigorous methods that ensure accuracy and reliability, and are less sensitive to the choice of basis set.

They are well-suited for dealing with singularities in spectral functions, and efficient algorithms are being developed to enable the study of larger and more complex systems. Realistic models of disorder are incorporated into the calculations, and the methods are being applied to a wide range of emerging 2D materials and heterostructures. This work is highly mathematical and computationally intensive, going beyond standard density functional theory calculations. The use of rigged Hilbert spaces, Chebyshev polynomials, and advanced numerical algorithms demonstrates a deep understanding of both physics and mathematics, and has the potential to significantly advance our understanding of 2D materials and heterostructures. The accurate and reliable methods developed could be used to design new materials with tailored electronic properties, predict material behaviour under different conditions, develop new electronic devices, and gain insights into fundamental physics. In conclusion, this is a comprehensive and sophisticated work that represents a significant contribution to the field of computational materials science, offering valuable insights into the development of advanced numerical methods for studying 2D materials.

High-Order Delta-Chebyshev Method for Spectral Densies

Scientists have developed a new numerical method for calculating spectral densities, achieving significantly improved accuracy in determining the local density of states (LDOS) of sparse Hamiltonians derived from tight-binding models. This work introduces the high-order delta-Chebyshev method, a refinement of the established regularized Chebyshev kernel polynomial method (KPM), and delivers substantial gains in convergence speed. The breakthrough lies in utilizing a high-order accurate approximation of the delta-function, enabling rapid convergence to the thermodynamic limit for smooth spectral densities. The team demonstrated that the computational demands of this new method are comparable to those of KPM, with the enhanced accuracy achieved through an inexpensive post-processing step.

Experiments applied the algorithm to tight-binding models of graphene and twisted bilayer graphene, confirming high-order convergence to the LDOS at non-singular points. Specifically, the method achieves O(p−m) pointwise convergence, where ‘p’ represents the expansion size and ‘m’ is an arbitrary positive integer parameter, representing a significant improvement over the O(p−2) convergence of standard KPM. Measurements confirm that the accuracy of the approximation is directly linked to the distance between the geometric truncation edge of the finite model and the site under consideration. This ensures that the spurious discrete spectrum is resolved, allowing for accurate calculation of the thermodynamic LDOS. The method’s success stems from a novel approach to approximating the delta-function, leveraging a technique previously used for infinite-dimensional operators, but adapted for large, sparse matrices common in tight-binding models. This allows scientists to avoid direct solution of linear systems, relying instead on efficient Chebyshev iteration.

High-Order Convergence for Local Density of States

The researchers developed a new numerical method, termed the high-order delta-Chebyshev method, for calculating the local density of states in complex physical systems, specifically sparse Hamiltonians derived from tight-binding models. This method builds upon the established regularized Chebyshev kernel polynomial method, but achieves significantly faster convergence to accurate results, particularly for smooth spectral densities. The key innovation lies in employing a high-order accurate approximation of the delta function, which enhances the precision of the calculations without substantially increasing computational cost. Demonstrating the method’s effectiveness, the team applied it to models of twisted bilayer graphene, successfully achieving high-order convergence to the local density of states at points where the density is well-behaved. Results indicate that the new method attains a higher level of accuracy than standard techniques for a given computational effort.