Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

From BCS-like superconductivity to condensation of local pairs: A numerical study of the attractive Hubbard model

Johannes Maria Singer, M. H. Pedersen, T. Schneider, H. Beck, Hans‐Georg Matuttis

Published Jul 1, 1996• 97 citations
Publisher

Abstract

We investigate the two-dimensional attractive Hubbard model with quantum Monte Carlo techniques to reveal the crossover from a BCS-type superconductivity in the weak-coupling regime to a superconductivity properly described by a Bose-Einstein condensation (BEC) of local, preformed pairs. The crossover from BCS to BEC is particularly well exposed in the temperature dependence of both the spin susceptibility and the double occupancy, as well as by the appearance of a pseudogap in the density of states far above ${\mathit{T}}_{\mathit{c}}$. These features are also mirrored in the shape of the specific-heat peak around ${\mathit{T}}_{\mathit{c}}$, the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1286

Related papers

Two-dimensional attractive Hubbard model and the BCS-BEC crossover

Rodrigo A. Fontenele, Natanael C. Costa, Raimundo R. dos Santos +1 more

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 road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature ${T}_{c}$ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength $U$ from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ ($t$ is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing ${T}_{p}$ and degeneracy ${T}_{d}$ 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), whereas 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.

Physical review. B./Physical review. B•May 2, 2022•34 citations

The 2D attractive Hubbard model and the BCS-BEC crossover

Rodrigo A. Fontenele, Natanael C. Costa, Raimundo R. dos Santos +1 more

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, $T_c$, in terms of the band filling, $\ave{n}$, and interaction strength, $U$, from which we pinpoint a somewhat wide region $|U|/t \approx 5 \pm 1$ ($t$ is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum $T_c \approx 0.16 t$. Two additional temperature scales, namely pairing, $T_p$, and degeneracy, $T_d$, 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.

arXiv:2201.02156•Jan 6, 2022•1 citation

Pairing Fluctuations and Anomalous Transport Above the BCS-BEC Crossover in the Two Dimensional Attractive Hubbard Model

Sabyasachi Tarat, Pinaki Majumdar

A Fermi liquid with weak attractive interaction undergoes a BCS transition to a superconductor with reducing temperature. With increasing interaction strength, the thermal transition is progressively modified as the high temperature `metallic' phase develops a pseudogap due to pairing fluctuations and the resistivity above T_c shows insulating behaviour. The crossover to insulating character occurs much before the system can be considered to be in the BEC regime of preformed fermion pairs. We use a new Monte Carlo tool to map out the BCS-BEC crossover in the attractive Hubbard model on large two dimensional lattices and explicitly compute the resistivity to demonstrate how the metal to superconductor (MS) thermal transition at weak coupling crosses over to an insulator to superconductor (IS) transition at intermediate coupling. Our high resolution access to the single particle and optical spectrum at finite temperature allows us to completely describe the transport crossover in this longstanding problem.

arXiv:1105.1156•May 5, 2011•0 citations

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

Hao Du, Qian Sun, Zhuotao Xie +1 more

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.

arXiv:2610.06319•Oct 5, 2026•0 citations

Pairing fluctuations and pseudogaps in the attractive Hubbard model

Bumsoo Kyung, S. Allen, A.–M. S. Tremblay

The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature ${T}^{*},$ also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around ${T}^{*}$ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.

arXiv:cond-mat/0010001•Physical review. B, Condensed matter•Jul 31, 2001•63 citations

Quantum Monte Carlo study of the three-dimensional attractive Hubbard model

Alain Sewer, X. Zotos, Hans Beck

We study the three-dimensional (3D) attractive Hubbard model by means of the determinant quantum Monte Carlo method. This model is a prototype for the description of the smooth crossover between BCS superconductivity and Bose-Einstein condensation. By detailed finite-size scaling we extract the finite-temperature phase diagram of the model. In particular, we interpret the observed behavior according to a scenario of two fundamental temperature scales; ${T}^{*}$ associated with Cooper pair formation and ${T}_{c}$ with condensation (giving rise to long-range superconducting order). Our results also indicate the presence of a recently conjectured phase transition hidden by the superconducting state. A comparison with the 2D case is briefly discussed, given its relevance for the physics of high-${T}_{c}$ cuprate superconductors.

arXiv:cond-mat/0204053•Physical review. B, Condensed matter•Oct 9, 2002•39 citations
We investigate the two-dimensional attractive Hubbard model with quantum Monte Carlo techniques to reveal the crossover from a BCS-type superconductivity in the weak-coupling regime to a superconductivity properly described by a Bose-Einstein condensation (BEC) of local, preformed pairs. The crossover from BCS to BEC is particularly well exposed in the temperature dependence of both the spin susceptibility and the double occupancy, as well as by the appearance of a pseudogap in the density of states far above Tc{\mathit{T}}_{\mathit{c}}Tc​. These features are also mirrored in the shape of the specific-heat peak around Tc{\mathit{T}}_{\mathit{c}}Tc​, the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.
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 road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature Tc{T}_{c}Tc​ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength UUU from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ (ttt is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing Tp{T}_{p}Tp​ and degeneracy Td{T}_{d}Td​ 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), whereas 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.
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.
A Fermi liquid with weak attractive interaction undergoes a BCS transition to a superconductor with reducing temperature. With increasing interaction strength, the thermal transition is progressively modified as the high temperature `metallic' phase develops a pseudogap due to pairing fluctuations and the resistivity above T_c shows insulating behaviour. The crossover to insulating character occurs much before the system can be considered to be in the BEC regime of preformed fermion pairs. We use a new Monte Carlo tool to map out the BCS-BEC crossover in the attractive Hubbard model on large two dimensional lattices and explicitly compute the resistivity to demonstrate how the metal to superconductor (MS) thermal transition at weak coupling crosses over to an insulator to superconductor (IS) transition at intermediate coupling. Our high resolution access to the single particle and optical spectrum at finite temperature allows us to completely describe the transport crossover in this longstanding problem.
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.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
We study the three-dimensional (3D) attractive Hubbard model by means of the determinant quantum Monte Carlo method. This model is a prototype for the description of the smooth crossover between BCS superconductivity and Bose-Einstein condensation. By detailed finite-size scaling we extract the finite-temperature phase diagram of the model. In particular, we interpret the observed behavior according to a scenario of two fundamental temperature scales; T∗{T}^{*}T∗ associated with Cooper pair formation and Tc{T}_{c}Tc​ with condensation (giving rise to long-range superconducting order). Our results also indicate the presence of a recently conjectured phase transition hidden by the superconducting state. A comparison with the 2D case is briefly discussed, given its relevance for the physics of high-Tc{T}_{c}Tc​ cuprate superconductors.