Crossover from Regular to Chaotic Behavior in the Conductance of Periodic Quantum Chains

TL;DR

This paper investigates how the conductance of a periodic quantum chain transitions from regular to chaotic behavior as impurity strength increases, revealing a crossover from staircase to band-structure influenced conductance.

Contribution

It introduces a quantitative measure of chaoticity in quantum chains and describes the conductance transition from regular to chaotic regimes based on impurity strength.

Findings

Conductance exhibits staircase behavior for few impurities.

For many impurities, conductance relates to the number of energy bands.

Conductance decreases from N to approximately N/2 as chaos increases.

Abstract

The conductance of a waveguide containing finite number of periodically placed identical point-like impurities is investigated. It has been calculated as a function of both the impurity strength and the number of impurities using the Landauer-B\"uttiker formula. In the case of few impurities the conductance is proportional to the number of the open channels $N$ of the empty waveguide and shows a regular staircase like behavior with step heights $\approx 2e^2/h$. For large number of impurities the influence of the band structure of the infinite periodic chain can be observed and the conductance is approximately the number of energy bands (smaller than $N$) times the universal constant $2e^2/h$. This lower value is reached exponentially with increasing number of impurities. As the strength of the impurity is increased the system passes from integrable to quantum-chaotic. The conductance, in units of $2e^2/h$, changes from $N$ corresponding to the empty waveguide to $\sim N/2 $ corresponding to chaotic or disordered system. It turnes out, that the conductance can be expressed as $(1-c/2)N$ where the parameter $0<c<1$ measures the chaoticity of the system.