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arXiv:cond-mat/9603163 · Physical review. B, Condensed matter

Low-density ferromagnetism in the Hubbard model

P. Pieri, S. Daul, D. Baeriswyl, Michael Dzierzawa, P. Fazekas

Published Oct 1, 1996• 29 citations
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Abstract

A single-band Hubbard model with nearest and next-nearest neighbor hopping is studied for d=1, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of U for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M\"uller-Hartmann. For three dimensions and nonpathological band structures, it is proven that such a scenario does not work. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.9250

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A single-band Hubbard model with nearest and next-nearest neighbor hopping is studied for d=1, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of U for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M"uller-Hartmann. For three dimensions and nonpathological band structures, it is proven that such a scenario does not work. \textcopyright{} 1996 The American Physical Society.
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We examined a one-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping with up to 32 sites by means of the density-matrix renormalization-group method. The model has two energy bands and includes the flat-band model studied by Tasaki as a special case. The model with finite band dispersion is found to show metallic (saturated) ferromagnetism for a sufficiently strong interaction. No partially polarized ground state is found. The interaction necessary to realize a ferromagnetic ground state becomes weak when the electron density is decreased from the quarter-filling.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
We study the existence of ferromagnetism in one dimensional Hubbard Model with on site interaction and nearest and next nearest neighbor hopping. Using the Hubbard I approximation, a self consistent equation for $$, the particle density with spin $σ$ is obtained. We find that ferromagnetism exists over a wide range of values of the next nearest neighbor hopping element t' when on-site interaction U is large. The phase diagram as a function of t'/t and the electron density $$ has been obtained by numerically solving the self consistent equation. It is found that the maximum density at which ferromagnetism can occur, peaks around t′/t≈−0.53t'/t \approx -0.53t′/t≈−0.53. We show that this phenomenon is a result of the peculiar way in which the t' term modifies the density of states.
Recent ultracold-atom experiments have observed ferromagnetic correlations in a geometrically tunable Fermi--Hubbard lattice at intermediate densities. Motivated by subsequent theoretical work that connected the low-density limit of the same model to the Mueller--Hartmann mechanism for itinerant ferromagnetism, we investigate the stability of ferromagnetism across a broad density range using the T-matrix approximation, a controlled approach in the dilute limit. We reproduce the ferromagnetic regime observed experimentally at intermediate densities and systematically compare finite-size and thermodynamic-limit calculations in the dilute regime. We find that the low-density ferromagnetic phase reported for the one-diagonal hopping lattice is strongly suppressed with increasing system size and disappears in the thermodynamic limit, indicating that it originates from finite-size effects. By contrast, low-density ferromagnetism remains robust in the square-lattice Hubbard model with hopping along both diagonal directions. We show that this qualitative difference cannot be explained by the degeneracy of the band minima alone, which occurs in both models. Instead, it is controlled by the singular behavior of the density of states at the band bottom: a weak divergence is insufficient to stabilize ferromagnetism, whereas a much stronger quasi-one-dimensional singularity supports a fully polarized ground state even in the dilute limit. Our results demonstrate that band-minimum degeneracy alone is not sufficient for low-density ferromagnetism and that the nature of the band-bottom density-of-states singularity ultimately controls its stability.