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Physical review. B, Condensed matter

Second-neighbor hopping in the attractive Hubbard model

Raimundo R. dos Santos

Published Sep 1, 1992• 11 citations
Publisher

Abstract

We study the attractive (i.e., negative-U) Hubbard model with nearest (t) and next-nearest-neighbor (${\mathit{t}}_{2}$) 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}, ${\mathit{t}}_{2}$ 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 ${\mathit{t}}_{2}$.

DOI
10.1103/physrevb.46.5496

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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​.
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.
The attractive Hubbard model plays a paradigmatic role in the study of superconductivity (superfluidity) and has become directly realizable in ultracold atom experiments on optical lattices. However, the critical temperatures, TcT_cTc​'s, remain lower than the lowest temperatures currently achievable in experiments. Here, we explore a possible route to enhance TcT_cTc​ by introducing an additional next-nearest-neighbor (NNN) hopping, t′t^\primet′, in a two-dimensional square lattice. We perform sign-problem-free determinant quantum Monte Carlo simulations to compute response functions such as pairing correlation functions, superfluid density, and uniform spin susceptibility. Our results show that a judicious choice of t′t^\primet′ can increase TcTcTc by up to 50%50\%50% compared to the case with only nearest-neighbor hopping. In contrast, the preformed pairs temperature scale, named pairing temperature, TpT_pTp​, decreases with increasing ∣t′/t∣|t^{\prime}/t|∣t′/t∣, which should represent a reduction of the pseudogap region, favoring a more BCS-like behavior at intermediate coupling. We further analyze the interacting density of states to characterize the transition from a pseudogap regime to a fully gapped superconducting state. These findings suggest that NNN hopping could be a viable route to increase TcT_cTc​ to values closer to experimentally accessible temperature scales.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (JJJ) and correlated hopping (KKK) interaction. Detailed numerical analysis of sss-, ppp-, and ddd-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (ttt) and next-nearest- (t2{t}_{2}t2​) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the ddd-wave and the extended sss-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the t2t\frac{{t}_{2}}{t}tt2​​ ratio, giving in particular for t2=0{t}_{2}=0t2​=0 sss-wave (ddd-wave) pairing as most favorable in the low (high) density limit, but increasing t2{t}_{2}t2​ can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects sss-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
The emergence of superconductivity (SC) in lattice models, such as the attractive Hubbard one, has renewed interest since the realization of cold-atom experiments. However, reducing the temperature in these experiments is a bottleneck; therefore, investigating how to increase the energy scale for SC is crucial to cold atoms. In view of this, we examine the effects of next-nearest-neighbor hoppings (t′) on the pairing properties of the attractive Hubbard model in a square lattice. To this end, we analyze the model through unbiased Quantum Monte Carlo simulations for fixed density n=0.87, and perform finite-size scaling analysis to the thermodynamic limit. As our main result, we notice that the existence of further hopping channels leads to an enhancement of the pairing correlations, which, in turn, increases the ground-state order parameter Δ. Finally, at finite temperatures, for t′/t≠0, this enhancement of pairing correlations leads to an increase in the critical temperature Tc. That is, the fine-tuning of second-neighbor hoppings increases the energy scales for SC, and may be a route by which cold-atom experiments can achieve such a phase and to help us further understand the nature of this phenomenon.
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.
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