Quantum Hall Effect

Imagine that a pair of counter-propagating flows (e.g., current, liquid) move oppositely and exchange associated information, such as voltage potential or pressure via some kind of interactions. In principle, the interaction can lead to a stationary condition that these information are only a function of locations and do not change with time. This stationary condition is not as trivial as that for a pair of co-propagating flows, which is the effective average of them. In this post, we will first focus on the interaction itself and lay down some basic algebra to describe this model, then move to the discussion about quantum Hall effect - a very rare and well-known macroscopic quantum phenomenon and the design of algorithms. We will wrap up this post with implementing algorithms in Python and introducing general usage of Python pacakge. ...

Background Landau level In solid-state physics, the famous band theory describes how electrons are distributed favorably over energy and momentum in electronic materials. The states (energy, momentum, spin and etc.) of electrons collectively form an electronic band structure. When subjecting to a magnetic field, these electronic states would shift in energy according to its interactive terms with the field. As this field gets very strong, the interactive term (orbital contribution) becomes dominant and start to form dense bands with narrow energy dispersion (A lot of states have almost the same energy). These bands are also energetically discrete and referred to as “Landau level”. ...