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Physical review. B, Condensed matter

Two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping

H. Q. Lin, J. E. Hirsch

Published Mar 1, 1987• 185 citations
Publisher

Abstract

We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (${t}_{2}$) 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 ${t}_{2}$\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.

DOI
10.1103/physrevb.35.3359

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Physical review. B, Condensed matter•Sep 1, 1992•11 citations
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.
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 ${t}^{\ensuremath{'}}$ 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=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 have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
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.
Abstract Using the strong coupling diagram technique, we investigate the extended Hubbard model on a two-dimensional square lattice. This approach allows for charge and spin fluctuations and a short-range antiferromagnetic order at nonzero temperatures. The model features the first-order phase transition to states with alternating site occupations (SAO) at the intersite repulsion v = v c . In this work, we show that doping decreases v c . For a nonzero next-nearest neighbor hopping t ′ , less mobile carriers produce a stronger fall in v c . For half-filling and t ′ = 0 , we consider phase diagrams for a fixed temperature T, on-site U, and intersite repulsions. The diagrams contain regions of SAO, Mott insulator, and several metallic states distinguished by their densities of states. Two of them are characterized by a dip and a peak at the Fermi level. The dip originates from the Slater mechanism for itinerant electrons, while the peak to bound states of electrons with localized magnetic moments. The existence of these two metallic regions in the phase diagram is a manifestation of the Pomeranchuk effect. The boundary between these regions reveals itself as a kink in the curve v c (T) and as a maximum in the T dependence of the double occupancy D. Our calculated D for different values of v, U, and T are in semiquantitative agreement with Monte Carlo results.
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.
We study the attractive (i.e., negative-U) Hubbard model with nearest (t) and next-nearest-neighbor (t2{\mathit{t}}_{2}t2​) hopping on a square lattice. A spin-down particle-hole transformation maps the negative-U model at any band filling onto a repulsive model with a magnetic field and at zero chemical potential; the sign of the second-neighbor hopping constant in this model depends on the spin state. For large \ensuremath{\Vert}U\ensuremath{\Vert}, t2{\mathit{t}}_{2}t2​ induces a coupling between spin operators in next-nearest-neighbor sites: antiferromagnetic between the z components and ferromagnetic along the transverse direction. The superconducting transition is investigated through Monte Carlo simulations on the attractive model. Fixing both U and the band filling, we find that the critical temperature increases steadily with t2{\mathit{t}}_{2}t2​.