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arXiv:cond-mat/9506083 · Physical review. B, Condensed matter

Influence of next-nearest-neighbor electron hopping on the static and dynamical properties of the two-dimensional Hubbard model

Daniel Duffy, Adriana Moreo

Published Dec 1, 1995• 67 citations
arXivHTMLPDF

Abstract

Comparing experimental data for high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors with numerical results for electronic models, it is becoming apparent that a hopping along the plaquette diagonals has to be included to obtain quantitative agreement. However, the values for ${\mathit{t}}^{\ensuremath{'}}$ discussed in the literature were obtained comparing theoretical results in the weak-coupling limit with photoemission data and band-structure calculations. The goal of this paper is to study how ${\mathit{t}}^{\ensuremath{'}}$ gets renormalized as the interaction between electrons, U, increases. For this purpose, the effect of adding a bare diagonal hopping ${\mathit{t}}^{\ensuremath{'}}$ to the two-dimensional Hubbard model Hamiltonian is investigated using numerical techniques. Spin-spin correlations, n(k), 〈n〉 vs \ensuremath{\mu}, and local magnetic moments are studied for several values of U/t and the electronic density. The spectral function A(k,\ensuremath{\omega}) is also discussed. We introduce a criterion to determine probable locations of Fermi surfaces at zero temperature. In general, we conclude that it is very dangerous to extract a bare parameter of the Hamiltonian (${\mathit{t}}^{\ensuremath{'}}$) from photoemission spectroscopy data where renormalized parameters play the important role.

DOI
10.1103/physrevb.52.15607

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Comparing experimental data for high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprate superconductors with numerical results for electronic models, it is becoming apparent that a hopping along the plaquette diagonals has to be included to obtain quantitative agreement. However, the values for t\ensuremath′{\mathit{t}}^{\ensuremath{'}}t\ensuremath′ discussed in the literature were obtained comparing theoretical results in the weak-coupling limit with photoemission data and band-structure calculations. The goal of this paper is to study how t\ensuremath′{\mathit{t}}^{\ensuremath{'}}t\ensuremath′ gets renormalized as the interaction between electrons, U, increases. For this purpose, the effect of adding a bare diagonal hopping t\ensuremath′{\mathit{t}}^{\ensuremath{'}}t\ensuremath′ to the two-dimensional Hubbard model Hamiltonian is investigated using numerical techniques. Spin-spin correlations, n(k), 〈n〉 vs \ensuremath{\mu}, and local magnetic moments are studied for several values of U/t and the electronic density. The spectral function A(k,\ensuremath{\omega}) is also discussed. We introduce a criterion to determine probable locations of Fermi surfaces at zero temperature. In general, we conclude that it is very dangerous to extract a bare parameter of the Hamiltonian (t\ensuremath′{\mathit{t}}^{\ensuremath{'}}t\ensuremath′) from photoemission spectroscopy data where renormalized parameters play the important role.
Comparing experimental data for high temperature cuprate superconductors with numerical results for electronic models, it is becoming apparent that a diagonal hopping has to be included to obtain quantitative agreement. According to recent estimations the value of the diagonal hopping, t{prime}, appears to be material dependent. However, the values for t{prime} discussed in the literature were obtained comparing theoretical results in the non-interacting and weak-coupling limit with experimental photoemission data. The goal of this paper is to study how t{prime} gets renormalized as the interaction between electrons, U, increases. For this purpose, the effect of adding a bare diagonal hopping t{prime} to the fully interacting two dimensional Hubbard model Hamiltonian is investigated using numerical techniques. Positive and negative values of t{prime} are analyzed. Magnetic correlations and incommensurability are studied for values of U/t ranging from 0 to 6, and as a function of the electronic density.
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.
Abstract Using the strong-coupling diagram technique, low-temperature spectral properties of the two-dimensional fermionic Hubbard model are considered for strong and moderate Hubbard repulsions U . The electron hopping to the nearest, second and third neighbors is taken into account with hopping constants t , t ′ = − 0.3 t and t ″ = 0.2 t , respectively. The nonzero values of t ′ and t ″ lead to strong asymmetry in magnetic properties with respect to the hole and electron doping—for U = 8 t strong antiferromagnetic correlations are retained up to the electron concentration n ¯ ≈ 1.25 , while they are destroyed completely at n ¯ ≈ 0.87 . When the temperature is decreased to T ≲ 0.1 t , in a wide range of electron concentrations there appear narrow and intensive peaks at the Fermi level (FL) in densities of states. For U ≲ 6 t the peaks are seen even at half-filling, while for larger U they arise as the FL leaves the Mott gap. The peaks are connected with a narrow band emerging at low temperatures. We identify states forming the band with spin-polaron excitations—bound states of correlated electrons and mobile spin excitations. Obtained low-temperature spectral functions are used for interpreting the peak-dip-hump structure observed in the photoemission of Nd 2− x Ce x CuO 4 . In the case of hole doping, the calculated Fermi contour contains arcs near nodal points with pseudogaps near antinodal points, while for electron doping the spectral intensity is suppressed at hot spots, in agreement with experimental observations in cuprate perovskites.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
Abstract Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping t′{t}^{{\prime} }t′ t′ or phonon energy Ω. Our results help elucidate how thee-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tccuprates.
The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism, ferromagnetism, superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations.