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arXiv:cond-mat/0703565 · Physical Review B

Hofstadter butterfly for a finite correlated system

Katarzyna Czajka, Anna Gorczyca, Maciej M. Maśka, Marcin Mierzejewski

Published Sep 27, 2006• 25 citations
arXivHTMLPDF

Abstract

We investigate a finite two-dimensional system in the presence of an external magnetic field. We discuss how the energy spectrum depends on the system size, boundary conditions, and Coulomb repulsion. On one hand, using these results, we present the field dependence of the transport properties of a nanosystem. In particular, we demonstrate that these properties depend on whether the system consists of an even or an odd number of sites. On the other hand, on the basis of exact results obtained for a finite system, we investigate whether the Hofstadter butterfly is robust against strong electronic correlations. We show that for sufficiently strong Coulomb repulsion, the Hubbard gap decreases when the magnetic field increases.

DOI
10.1103/physrevb.74.125116

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We investigate a finite two-dimensional system in the presence of an external magnetic field. We discuss how the energy spectrum depends on the system size, boundary conditions, and Coulomb repulsion. On one hand, using these results, we present the field dependence of the transport properties of a nanosystem. In particular, we demonstrate that these properties depend on whether the system consists of an even or an odd number of sites. On the other hand, on the basis of exact results obtained for a finite system, we investigate whether the Hofstadter butterfly is robust against strong electronic correlations. We show that for sufficiently strong Coulomb repulsion, the Hubbard gap decreases when the magnetic field increases.
We study thermodynamic properties of the two-dimensional (2D) Falicov-Kimball model in the presence of external magnetic field perpendicular to the lattice.The field is taken into account by the Peierls substitution in the hopping term.We show how the Hofstadter butterfly is affected by electronic correlations.In the non-interacting case the field dependent energy spectrum forms the famous Hofstadter butterfly.Our results indicate that for arbitrary nonzero interaction strength and arbitrary magnetic field there is a gap in the energy spectrum at sufficiently low temperature.The gap vanishes with increase of temperature for weak coupling, however, it persists at high temperatures if the coupling is strong enough.Numerical results have been obtained with the help of Monte Carlo technique based on a modified Metropolis algorithm.
Abstract The Hofstadter–Hubbard model captures the physics of strongly correlated electrons in an applied magnetic field, which is relevant to many recent experiments on Moiré materials. Few large-scale, numerically exact simulations exists for this model. In this work, we simulate the Hubbard–Hofstadter model using the determinant quantum Monte Carlo (DQMC) algorithm. We report the field and Hubbard interaction strength dependence of charge compressibility, fermion sign, local moment, magnetic structure factor, and specific heat. The gross structure of magnetic Bloch bands and band gaps determined by the non-interacting Hofstadter spectrum is preserved in the presence of U. Incompressible regions of the phase diagram have improved fermion sign. At half filling and intermediate and larger couplings, a strong orbital magnetic field delocalizes electrons and reduces the effect of Hubbard U on thermodynamic properties of the system.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
We analyze the two-body spectrum within the Hofstadter-Hubbard model on a square lattice through an exact variational ansatz and study the topological properties of its low-lying two-body bound-state branches. In particular, we discuss how the Hofstadter-Hubbard butterfly of the two-body branches evolves as a function of onsite interactions and how to efficiently calculate their Chern numbers using the Fukui-Hatsugai-Suzuki approach. Our numerical results are fully consistent with the simple picture that appears in the strong-coupling limit, where the attraction between fermions forms a composite boson characterized by an effective hopping parameter and an effective magnetic-flux ratio.
We study the revival of Hofstadter butterfly due to the competition between disorder and electronic interaction using mean field approximation of unrestricted Hartree Fock method at zero temperature for two dimensional square and honeycomb lattices. Interplay of disorder and electronic correlation to nullify each other is corroborated by the fact that honeycomb lattice needs more strength of electronic correlation owing to its less co-ordination number which enhances the effect of disorder. The extent of revival of the butterfly is better in square lattice than honeycomb lattice due to higher coordination number. The effect of disorder and interaction is also investigated to study entanglement entropy and entanglement spectrum. We find that for honeycomb lattice area law of entanglement entropy is obeyed in all cases but for square lattice there is some departure from area law for larger subsystems. The entanglement spectrum have the reflection symmetry of the original butterfly of the Hofstadter spectrum. The interaction induces a gap in the entanglement spectrum as well conforming the correspondence between physical spectrum and entanglement spectrum. The effect of disorder closes the interaction induced gap in the entanglement spectrum establishing the nullification of interaction due to disorder and vice versa.
We present exact analytical results for a critical field and correlations functions for N site ring and arbitrary size two dimensional coupled chains in a two component Hubbard model with a strong magnetic field and exactly at half-filling. The critical field near the Nagaoka instability with one spin and two spin flips is driven by Coulomb interaction, lattice geometry, dimensionality and crucially depends on the number of atomic sites N. Unsaturated ferromagnetism, reminiscent of Nagaoka-like behavior, and spin liquid state are examined in the structure of off-diagonal spin correlation at large U.