Propagators in two-dimensional lattices

TL;DR

This paper develops methods to compute discrete propagators in two-dimensional crystal lattices, applying tight-binding models and special functions, and explores their applications to diffusive phenomena and various lattice geometries.

Contribution

It introduces a unified approach to calculating propagators in 2D lattices using Dirac matrices and Bessel functions, covering multiple lattice types and diffusive effects.

Findings

Derived explicit propagator formulas for square, triangular, and hexagonal lattices.

Analyzed diffusive behavior of point distributions on crystalline sheets.

Provided a comprehensive compendium of propagators for different lattice geometries.

Abstract

This paper is devoted to the computation of discrete propagators in two-dimensional crystals and their application to a number of time dependent problems. The methods to compute such kernels are provided by a tight-binding representation of Dirac matrices and the generalizations of Bessel functions. Diffusive effects of point-like distributions on crystalline sheets are studied in a second quantization scheme. In the last part, a compendium of propagators is presented. The cases of square, triangular and hexagonal arrays are covered.