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arXiv:2503.06753 · Physical review. B./Physical review. B

Effects of next-nearest neighbor hopping on the pairing and critical temperatures of the attractive Hubbard model on a square lattice

Rodrigo A. Fontenele, Natanael C. Costa, Thereza Paiva, Raimundo R. dos Santos

Published Mar 3, 2026• 1 citation
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Abstract

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, $T_c$'s, remain lower than the lowest temperatures currently achievable in experiments. Here, we explore a possible route to enhance $T_c$ by introducing an additional next-nearest-neighbor (NNN) hopping, $t^\prime$, 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 $t^\prime$ can increase $Tc$ by up to $50\%$ compared to the case with only nearest-neighbor hopping. In contrast, the preformed pairs temperature scale, named pairing temperature, $T_p$, decreases with increasing $|t^{\prime}/t|$, 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 $T_c$ to values closer to experimentally accessible temperature scales.

DOI
10.1103/tgs1-r698

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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.
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.
The attractive Hubbard model has become a model readily realizable with ultracold atoms on optical lattices. However, the superconducting (superfluid) critical temperatures Tc{T}_{c}Tc​'s are still somewhat smaller than the lowest temperatures achieved in experiments. Here, we consider two possible routes, generically called layering, to increase Tc{T}_{c}Tc​: a bilayer and a simple cubic lattice, both with tunable hopping tz{t}_{z}tz​ between attractive Hubbard planes. We have performed minus-sign--free determinant quantum Monte Carlo simulations to calculate response functions such as pairing correlation functions, uniform spin susceptibility, and double occupancy, through which we map out some physical properties. We have found that by a judicious choice of fillings and intensity of on-site attraction, a bilayer can exhibit Tc{T}_{c}Tc​'s between 1.5 and 1.7 times those of the single layer; for the simple-cubic lattice the enhancement can be 30% larger than the maximum for the single layer. We also check the accuracy of both a BCS-like estimate for Tc{T}_{c}Tc​ in the attractive Hubbard model as well as of an upper bound for Tc{T}_{c}Tc​ based on the superfluid density.
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​.
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.
In the half-filled one-orbital Hubbard model on a square lattice, we study the effect of next-nearest neighbor hopping on the single-particle spectral function at finite temperature using an exact-diagonalization + Monte-Carlo based approach to the simulation process. We find that the pseudogap-like dip, existing in the density of states in between the Néel temperature TNT_NTN​ and a relatively higher temperature T∗T^*T∗, is accompanied with a significant asymmetry in the hole- and particle-excitation energy along the high-symmetry directions as well as along the normal-state Fermi surface. On moving from (π/2,π/2π/2, π/2π/2,π/2) toward (π,0)(π, 0)(π,0) along the normal state Fermi surface, the hole-excitation energy increases, a behavior remarkably similar to what is observed in the ddd-wave state and pseudogap phase of high-TcT_cTc​ cuprates, whereas the particle-excitation energy decreases. The quasiparticle peak height is the largest near (π/2,π/2π/2, π/2π/2,π/2) whereas it is the smallest near (π,0)(π, 0)(π,0). These spectral features survive beyond TNT_NTN​. The temperature window TN≲T≲T∗T_N \lesssim T \lesssim T^*TN​≲T≲T∗ shrinks with an increase in the next-nearest neighbor hopping, which indicates that the next-nearest neighbor hopping may not be supportive to the pseudogap-like features.
In the half-filled one-orbital Hubbard model on a square lattice, we study the effect of next-nearest neighbor hopping on the single-particle spectral function at finite temperature using an exact-diagonalization + Monte-Carlo based approach to the simulation process. We find that the pseudogap-like dip, existing in the density of states in between the Néel temperature T_N T N and a relatively higher temperature scale T^* T * , is accompanied with a significant asymmetry in the hole- and particle-excitation energy along the high-symmetry directions as well as along the normal-state Fermi surface. On moving from ( \pi/2, \pi/2 π / 2 , π / 2 ) toward (\pi, 0) ( π , 0 ) along the normal state Fermi surface, the hole-excitation energy increases, a behavior remarkably similar to what is observed in the d d -wave state and pseudogap phase of high- T_c T c cuprates, whereas the particle-excitation energy decreases. The quasiparticle peak height is the largest near ( \pi/2, \pi/2 π / 2 , π / 2 ) whereas it is the smallest near (\pi, 0) ( π , 0 ) . These spectral features survive beyond T_N T N . The temperature window T_N ≲ T ≲ T^* T N ≲ T ≲ T * shrinks with an increase in the next-nearest neighbor hopping, which indicates that the next-nearest neighbor hopping may not be supportive to the pseudogap-like features.