Study of Correlated Disorders and interaction in the Hofstadter Butterfly

TL;DR

This work investigates how correlated quasi-periodic disorders and electron–electron interactions affect the Hofstadter butterfly on a 2D square lattice. Using a zero-temperature mean-field Hubbard model in a magnetic field, it analyzes four deterministic sequences (RS, Fibonacci, Thue–Morse, AA) and interpolations between AA and each disorder to map changes in the spectrum, entanglement, and localization. Key findings are that AA disorder produces the strongest spectral restructuring and central gaps, strong disorder can erase the butterfly, while interactions can induce a partial revival of fractal features; entanglement follows area law at extreme fields but shows pronounced intermediate-field deviations, and IPR/NPR track increasing localization with disorder, with AA yielding the largest IPR. The results illuminate the delicate interplay between quasiperiodic order, interactions, and magnetic fields in fractal spectra, offering insights for correlated quasiperiodic materials and engineered lattices where disorder and interactions coexist.

Abstract

We investigate the impact of several quasiperiodic disorders and their continuous interpolation with the Aubry-Andre (AA) potential on the Hofstadter butterfly using mean field approximation at zero temperature for a two-dimensional square lattice. Weak disorder mildly smears the fractal spectrum, while strong quasiperiodic potentials destroy the butterfly and generate multiple energy gaps. The AA potential produces the strongest spectral restructuring, creating prominent gaps near half-filling. Interpolating AA with other quasiperiodic potentials reveals competing gap-opening mechanisms, ranging from AA-dominated gaps at small interpolation parameters to a robust half-filling gap generated by the competing disorders at large parameters. Entanglement entropy follows the area law at low and high magnetic fields but shows pronounced deviations at intermediate fields, with opposite trends for strong AA versus other quasiperiodic potentials. Localization analysis using IPR and NPR confirms enhanced localization with increasing disorder; the AA potential yields the largest IPR, with notable field dependence. Interpolation produces smooth crossovers between distinct localization regimes.

Figures (20)

• Figure 1: Schematic of the arrangement of the two values for the periodic and the deterministic aperiodic sequences.
• Figure 2: The gap at half filling as a function of coulomb interaction parameter $U$ , is shown for four different corelated disorders.
• Figure 3: The energy spectrum (E) is plotted as a function of the magnetic field (B) in a square lattice, illustrating the effects of various types of disorders and different combinations of disorder and interaction. Each plot is labeled to represent a specific configuration of disorder and interaction. As seen from the spectrum, all types of disorder tend to smear out the characteristic butterfly structure. The on-site interaction alone opens a clear gap in the spectrum Mandal, whereas the simultaneous presence of disorder and interaction closes this gap and, to some extent, restores features of the original butterfly pattern.
• Figure 4: The energy spectrum (E) is plotted as a function of the magnetic field (B) in a square lattice, illustrating the effects of various types of disorder for two values of their strength $d=2,5$. Each plot is labeled to represent a specific configuration of disorders and their strength.
• Figure 5: Energy spectrum $E$ as a function of the magnetic field $B$ for a square lattice, showing the combined effects of the Aubry--André potential and other quasiperiodic disorders. The upper panel (a--d) illustrates the competition between AA and Thue--Morse disorders: the system evolves from the AA-dominated regime, characterized by two symmetric gaps, to the TM-dominated regime, where a single central gap appears. The lower panel displays the corresponding interpolation between the AA and Fibonacci disorders.