Infinite chain of N different deltas: a simple model for a Quantum Wire

TL;DR

This paper introduces an exact diagonalization method for a periodic potential with N different delta functions, modeling quantum wire band structures and revealing phenomena like Anderson localization and fractality in conductance.

Contribution

It provides an exact analytical approach for N-delta potentials in infinite chains, capturing key quantum wire properties and localization effects.

Findings

Exact band structure calculations for infinite chains

Observation of Anderson localization in random chains

Detection of fractal patterns in conductance

Abstract

We present the exact diagonalization of the Schrodinger operator corresponding to a periodic potential with N deltas of different couplings, for arbitrary N. This basic structure can repeat itself an infinite number of times. Calculations of band structure can be performed with a high degree of accuracy for an infinite chain and of the correspondent eigenlevels in the case of a random chain. The main physical motivation is to modelate quantum wire band structure and the calculation of the associated density of states. These quantities show the fundamental properties we expect for periodic structures although for low energy the band gaps follow unpredictable patterns. In the case of random chains we find Anderson localization; we analize also the role of the eigenstates in the localization patterns and find clear signals of fractality in the conductance. In spite of the simplicity of the model many of the salient features expected in a quantum wire are well reproduced.