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arXiv:cond-mat/9609253 · Physical Review Letters

Ferromagnetism in the Two Dimensional t − t ′ Hubbard Model at the Van Hove Density

R. Hlubina, Sandro Sorella, F. Guinea

Published Feb 17, 1997• 138 citations
arXivHTMLPDF

Abstract

Using an improved version of the projection quantum Monte Carlo technique, we study the square-lattice Hubbard model with nearest-neighbor hopping $t$ and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ by simulation of lattices with up to $20\ifmmode\times\else\texttimes\fi{}20$ sites. For a given ${R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2t}^{\ensuremath{'}}/t$, we consider that filling which leads to a singular density of states of the noninteracting problem. For repulsive interactions, we find an itinerant ferromagnet (antiferromagnet) for $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.94$ ( $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2$). This is consistent with the prediction of the $T$-matrix approximation, which sums the most singular set of diagrams.

DOI
10.1103/physrevlett.78.1343

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Using an improved version of the projection quantum Monte Carlo technique, we study the square-lattice Hubbard model with nearest-neighbor hopping ttt and next-nearest-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ by simulation of lattices with up to 20\ifmmode×\else\texttimes\fi2020\ifmmode\times\else\texttimes\fi{}2020\ifmmode×\else\texttimes\fi20 sites. For a given R=2t\ensuremath′/t{R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2t}^{\ensuremath{'}}/tR=2t\ensuremath′/t, we consider that filling which leads to a singular density of states of the noninteracting problem. For repulsive interactions, we find an itinerant ferromagnet (antiferromagnet) for R=0.94R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.94R=0.94 ( R=0.2R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2R=0.2). This is consistent with the prediction of the TTT-matrix approximation, which sums the most singular set of diagrams.
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.
A combination of analytical approaches and quantum Monte Carlo simulations is used to study both magnetic and pairing correlations for a version of the Hubbard model that includes second-neighbor hopping ${\mathit{t}}^{\ensuremath{'}}$=-0.35t as a model for high-temperature superconductors. Magnetic properties are analyzed using the two-particle self-consistent approach. The maximum in magnetic susceptibility as a function of doping appears both at finite ${\mathit{t}}^{\ensuremath{'}}$ and at ${\mathit{t}}^{\ensuremath{'}}$=0 but for two totally different physical reasons. When ${\mathit{t}}^{\ensuremath{'}}$=0, it is induced by antiferromagnetic correlations while at ${\mathit{t}}^{\ensuremath{'}}$=-0.35t it is a band structure effect amplified by interactions. Finally, pairing fluctuations are compared with T-matrix results to disentangle the effects of van Hove singularity and of nesting on superconducting correlations. The addition of antiferromagnetic fluctuations increases slightly the d-wave superconducting correlations despite the presence of a van Hove singularity which tends to decrease them in the repulsive model. Some aspects of the phase diagram and some subtleties of finite-size scaling in Monte Carlo simulations, such as inverted finite-size dependence, are also discussed.
Abstract We investigate magnetic correlation in the triangular lattice Hubbard model using determinant quantum Monte Carlo simulations. Focusing on the role of next-nearest-neighbor hopping t ′ and electron filling 〈 n 〉, we demonstrate that regions of high density of states, particularly near van Hove singularity (VHS) points, significantly enhance short-range ferromagnetic correlations as measured by the uniform spin susceptibility χ (Γ). Specifically, χ (Γ), quantifying ferromagnetic fluctuations, is amplified at fillings corresponding to the VHS for a given t ′ . Increasing the on-site Coulomb repulsion U further strengthens these ferromagnetic correlations, with a more pronounced effect at lower temperatures, and the observed ferromagnetic correlations are found to be short-ranged. Conversely, near half-filling 〈 n 〉 = 1.0, larger t ′ values promote antiferromagnetic (AFM) fluctuations, evidenced by an increase in χ ( K ). Our results reveal that t ′ and filling act as effective tuning parameters for manipulating the competition between ferromagnetic and AFM fluctuations in the triangular lattice.
Using exact diagonalization techniques, the Hubbard model with hopping between nearest (t) and next-nearest neighbors ${(t}^{\ensuremath{'}})$ is studied in a two-dimensional cluster with $4\ifmmode\times\else\texttimes\fi{}4$ sites. Within a range of values of ${t}^{\ensuremath{'}}<0$ and low densities, states with different total spin are degenerate in the noninteracting system. At a finite value of the Coulomb repulsion U, states with high spin are energetically favored and the ground state can be fully polarized. The behavior of the density of states differs from that obtained within the Hartree-Fock approximation. The low energy region of the density of states (DOS) for the majority spins remains weakly modified by U. Our results bring further support to the idea that itinerant ferromagnetism occurs in strongly correlated systems with high noninteracting DOS at the bottom of the band.
We investigate magnetic and superconducting instabilities of the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model on a square lattice at van Hove densities from weak to intermediate coupling by means of the two-particle self-consistent approach. We find that as the next-nearest-neighbor hopping $|{t}^{\ensuremath{'}}|$ increases from zero, the leading instability is towards an incommensurate spin-density wave whose wave vector moves slowly away from $(\ensuremath{\pi},\ensuremath{\pi}).$ For intermediate values of $|{t}^{\ensuremath{'}}|,$ the leading instability is towards ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity. For larger $|{t}^{\ensuremath{'}}|>0.33t,$ there are signs of a crossover to ferromagnetism at extremely low temperatures. The suppression of the crossover temperature is driven by Kanamori screening that strongly renormalizes the effective interaction and also causes the crossover temperature to depend only weakly on ${t}^{\ensuremath{'}}.$ Electronic self-energy effects for large $|{t}^{\ensuremath{'}}|$ lead to considerable reduction of the zero-energy single-particle spectral weight beginning at temperatures as high as $T\ensuremath{\lesssim}0.1t,$ an effect that may be detrimental to the existence of a ferromagnetic ground state at weak coupling.
The magnetic susceptibility of the two-dimensional repulsive Hubbard model with nearest-neighbor hopping is investigated using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. At half-filling, the calculated zero-frequency susceptibility and the square of the site spin reproduce adequately the results of Monte Carlo simulations. Also, in agreement with numerical simulations, no evidence of ferromagnetic correlations was found in the considered range of electron concentrations $0.8\ensuremath{\lesssim}\overline{n}\ensuremath{\lesssim}1.2$ for the repulsion parameters $8\ensuremath{\mid}t\ensuremath{\mid}\ensuremath{\leqslant}U\ensuremath{\leqslant}16\ensuremath{\mid}t\ensuremath{\mid}$. However, for larger $U∕\ensuremath{\mid}t\ensuremath{\mid}$ and $\ensuremath{\mid}1\ensuremath{-}\overline{n}\ensuremath{\mid}\ensuremath{\approx}0.2$, the nearest-neighbor correlations become ferromagnetic. For $\overline{n}\ensuremath{\lesssim}0.94$ and $\overline{n}\ensuremath{\gtrsim}1.06$, the imaginary part of the real-frequency susceptibility becomes incommensurate for small frequencies. The incommensurability parameter grows with departure from half-filling and decreases with increasing frequency. This behavior of the susceptibility can explain the observed low-frequency incommensurate response observed in normal-state lanthanum cuprates.