The integer quantum Hall effect of a square lattice with an array of point defects

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Solid-State Physics

Background:

• The Hofstadter butterfly describes the energy spectrum of electrons in a 2D lattice under a magnetic field.
• Impurities and vacancies can significantly alter electronic properties, leading to complex phenomena.

Purpose of the Study:

• To investigate the electronic properties of a square lattice with impurities/vacancies under a perpendicular magnetic field.
• To analyze the impact of second-nearest neighbor interactions on the Hofstadter butterfly spectrum.
• To determine the Hall conduction spectrum in the presence of these imperfections.

Main Methods:

• Tight-binding method including up to second nearest neighbor interactions.
• Kubo formula for calculating the Hall conduction spectrum.
• Analysis of spectral properties and localization effects.

Main Results:

• Impurities and vacancies introduce new gaps and bands into the Hofstadter butterfly, even when bipartite symmetry is broken.
• Localized states from low hopping constant impurities do not contribute to Hall conduction.
• Impurities with high hopping constants lead to new Hall plateaus with quantized conduction (σ(xy) = ±e²/h).

Conclusions:

• Electronic imperfections significantly modify the Hofstadter butterfly and Hall conduction.
• The nature of impurity hopping constants dictates their contribution to conductivity.
• The findings align with the Thouless-Kohmoto-Nightingale-den Nijs integers for Fermi energies within energy gaps.