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arXiv:1308.0664 · Journal of the Physical Society of Japan

BCS–BEC Crossover in the Two-Dimensional Attractive Hubbard Model: Variational Cluster Approach

Tatsuya Kaneko, Yukinori Ohta

Published Jan 21, 2014• 9 citations
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Abstract

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 $ξ$ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state ($ξ\gg a$) to a BEC state of tightly bound pairs ($ξ\ll a$), where $a$ 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.

DOI
10.7566/jpsj.83.024711

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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.
The BCS-BEC crossover within the two-dimensional attractive Hubbard model is studied by using the Cellular Dynamical Mean-Field Theory, both in the normal and superconducting ground states. Short-range spatial correlations incorporated in this theory remove the normal-state quasiparticle peak and the first-order transition found in the Dynamical Mean-Field Theory, rendering the normal state crossover smooth. For UUU smaller than the bandwidth, pairing is driven by the potential energy, while in the opposite case it is driven by the kinetic energy, resembling a recent optical conductivity experiment in cuprates. Phase coherence leads to the appearance of a collective Bogoliubov mode in the density-density correlation function and to the sharpening of the spectral function.
The superfluid state in the attractive Hubbard is studied for the stacked square lattices to simulate the two- to three-dimensional crossover effects. We use the variation cluster approximation to calculate the order parameter and pair coherence length from the weak-coupling BCS state to the strong-coupling BEC state at T = 0 K. We show that in the weak-coupling BCS region the density of states at Fermi level plays an important role in the dimensional crossover of the superfluid state but in the strong-coupling BEC region the quantum fluctuations depending on the dimensionality of the system play an essential role.
In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).
We study properties of normal, superconducting (SC) and CDW states for an attractive Hubbard model on the square lattice, using a variational Monte Carlo method. In trial wave functions, we introduce an interspinon binding factor, indispensable to induce a spin-gap transition in the normal state, in addition to the onsite attractive and intersite repulsive factors. It is found that, in the normal state, as the interaction strength ∣U∣/t|U|/t∣U∣/t increases, a first-order spin-gap transition arises at ∣Uc∣∼W|U_{\rm c}|\sim W∣Uc​∣∼W (WWW: band width) from a Fermi liquid to a spin-gapped state, which is conductive through hopping of doublons. In the SC state, we confirm by analysis of various quantities that the mechanism of superconductivity undergoes a smooth crossover at around $|U_{\ma{co}}|\sim |U_{\rm c}|$ from a BCS type to a Bose-Einstein condensation (BEC) type, as ∣U∣/t|U|/t∣U∣/t increases. For $|U||U_{\ma{co}}|$, quantities such as the energy gain in the SC transition and superfluid stiffness, which is related to the cost of phase coherence, behave as ∼t2/∣U∣∝Tc\sim t^2/|U|\propto T_{\rm c}∼t2/∣U∣∝Tc​, as expected in a bosonic scheme. In this regime, the SC transition is induced by a gain in kinetic energy, in contrast with the BCS theory. We refer to the relevance to the pseudogap in 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 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 investigate electron-hole pair condensation in electron bilayers described by the square-lattice extended Falicov-Kimball model. Using exact diagonalization and variational cluster approximation techniques we first calculate the anomalous Green’s function to clarify the character of the excitons in momentum space. We then evaluate the coherence length ξ (in unit of the lattice constant a) from the corresponding condensation amplitude and demonstrate the smooth crossover between a BCS state of weakly paired electrons and holes (ξ/a 1) and a BEC state of tightly bound excitons (ξ/a 1) as the Coulomb attraction increases. Overcoming the finite-size effects of exact diagonalization while still taking into account the essential correlation effects we show that the variational cluster approximation provides an advantageous description of exciton condensation in strongly correlated electron systems.