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arXiv:cond-mat/0204053 · Physical review. B, Condensed matter

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

Alain Sewer, X. Zotos, Hans Beck

Published Oct 9, 2002• 39 citations
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Abstract

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.

DOI
10.1103/physrevb.66.140504

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arXiv:cond-mat/9405078•May 26, 1994•0 citations
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.
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.
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.
We study the phase diagram for the attractive (i.e., negative-U) Hubbard model on a simple-cubic lattice, through Monte Carlo simulations. We obtain the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ for superconductivity from a finite-size scaling analysis of the data for the pairing correlations. For fixed on-site attraction U, Tc{\mathit{T}}_{\mathit{c}}Tc​ displays a maximum near the filling factor 0.9, roughly independent of U. For fixed filling we estimate the crossover temperature ${\mathit{T}}^{\ifmmode\times\else\texttimes\fi{}}$(U), separating the normal states: metallic and spin gap. There is also a critical value Up{\mathit{U}}_{\mathit{p}}Up​ for pair formation, the magnitude of which seems to be independent of doping. The relevance of these results to the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ oxides is discussed.
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.
We consider a lattice version of the ``negative-U′′U''U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum XY\mathrm{XY}XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group E2),{E}_{2}),E2​), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature Tc{T}_{c}Tc​ as a function of the coupling strength ∣U∣,|U|,∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature Tg{T}_{g}Tg​ due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength ∣U∣/t,|U|/t,∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing ∣U∣|U|∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.
We study the phase diagram for the attractive ({\it i.e.,} negative-UUU) Hubbard model on a simple cubic lattice, through Monte Carlo simulations. We obtain the critical temperature, TcT_cTc​, for superconductivity from a finite-size scaling analysis of the data for the pairing correlations. For fixed on-site attraction, UUU, TcT_cTc​ displays a maximum near the filling factor 0.9, roughly independent of UUU. For fixed filling we estimate the crossover temperature T×(U)T^\times(U)T×(U), separating the normal states: metallic and spin-gap. There is also a critical value UpU_pUp​ for pair formation, the magnitude of which seems to be independent of doping. The relevance of these results to the high-TcT_cTc​ oxides is discussed.