Hofstadter butterfly in the Falicov-Kimball model on some finite 2D lattices

TL;DR

This study explores the Hofstadter butterfly spectrum in the Falicov-Kimball model on finite 2D lattices, revealing how magnetic fields, electron interactions, and finite size influence spectral gaps and orbital currents.

Contribution

It provides new insights into the effects of strong correlations and finite size on the Hofstadter spectrum in triangular and square lattices using exact diagonalization.

Findings

Magnetic field can induce a spectral gap without electron correlation.

Finite size introduces additional states within the spectral gap.

Electronic correlations can suppress extra states in the Hofstadter spectrum.

Abstract

Spinless, interacting electrons on a finite size triangular lattice moving in an extremely strong perpendicular magnetic field are studied and compared with the results on a square lattice. Using a Falicov-Kimball model, the effects of the magnetic field, Coulomb correlation and finite system size on their energy spectrum are observed. It is possible to induce a gap in the spectrum by tuning the magnetic field even in the absence of correlation, though extra states appear in the gap due to finite size. An orbital current is calculated for both the square and triangular lattice with and without electron correlation. In the noninteracting limit, the bulk current shows several patterns, while the edge current shows oscillations with magnetic flux. The oscillations persist in the interacting limit for the square lattice but not for the triangular lattice. Using exact diagonalization techniques, the recursive structure of Hofstadter spectrum is examined with strong electronic correlations for different system sizes. Electronic correlation is found to suppress these extra states in the gap in some cases.