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arXiv:2610.06319

Pairing order and condensate fraction across the BCS-BEC crossover in the two-dimensional attractive Hubbard model

Hao Du, Qian Sun, Zhuotao Xie, Yuan-Yao He

Published Oct 5, 2026• 0 citations
arXivHTMLPDF

Abstract

We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength $U/t=2\sim 12$ (where $t$ is the nearest-neighbor hopping amplitude) and fermion density $n=0.25\sim 1.0$. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.

DOI
10.48550/arxiv.2610.06319

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We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength U/t=2∼12U/t=2\sim 12U/t=2∼12 (where ttt is the nearest-neighbor hopping amplitude) and fermion density n=0.25∼1.0n=0.25\sim 1.0n=0.25∼1.0. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.
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Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, TcT_cTc​, in terms of the band filling, $\ave{n}$, and interaction strength, UUU, from which we pinpoint a somewhat wide region ∣U∣/t≈5±1|U|/t \approx 5 \pm 1∣U∣/t≈5±1 (ttt is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum Tc≈0.16tT_c \approx 0.16 tTc​≈0.16t. Two additional temperature scales, namely pairing, TpT_pTp​, and degeneracy, TdT_dTd​, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.
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.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
We present an effective Hamiltonian based real-space approach for studying the weak-coupling BCS to the strong-coupling Bose-Einstein condensate crossover in the two-dimensional attractive Hubbard model at finite temperatures. We introduce and justify an effective classical Hamiltonian to describe the thermal fluctuations of the relevant auxiliary fields. Our results for Tc{T}_{c}Tc​ and phase diagrams compare very well with those obtained from more sophisticated and CPU-intensive numerical methods. We demonstrate that the method works in the presence of disorder and can be a powerful tool for a real-space description of the effect of disorder on superconductivity. From a combined analysis of the superconducting order parameter, the distribution of auxiliary fields, and the quasiparticle density of states, we identify the regions of metallic, insulating, superconducting, and pseudogapped behavior. Our finding of the importance of phase fluctuations for the pseudogap behavior is consistent with the conclusions drawn from recent experiments on NbN superconductors. The method can be generalized to study superconductors with nontrivial order-parameter symmetries by identifying the relevant auxiliary variables.
We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length ξξξ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state (ξ≫aξ\gg aξ≫a) to a BEC state of tightly bound pairs (ξ≪aξ\ll aξ≪a), where aaa is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.