A one-to-two dimensional mapping using a partial fast Fourier transform

TL;DR

This paper presents a novel method to map a one-dimensional tight binding model to a two-dimensional model using a partial FFT, enabling exact transformations with potential applications in condensed matter physics.

Contribution

It introduces a partial FFT-based mapping technique that exactly transforms a 1D tight binding model into a 2D model with flux, a novel approach in lattice model analysis.

Findings

Exact 1D to 2D mapping achieved via partial FFT steps

Recursive decoupling into sublattices of half the size

Potential applications in modeling quantum systems

Abstract

It will be shown how to map a simple one-dimensional tight binding model with a cosine potential in one dimension exactly to a two dimensional tight binding model with periodic boundary conditions with the presence of a single flux quantum spread evenly on the torus. The mapping is is achieved by a partial sequence of "Fast Fourier Transform" (FFT) steps which if completed would be an exact Fourier transform of the original model. Each step of the FFT recursively maps a tight binding model into two decoupled sublattices of half the lattice length.