We point out the curious phenomenon of order by projection in a class of lattice Fermi systems near half filling. Enhanced pairing correlations of extended s-wave Cooper pairs result from the process of projecting out s-wave Cooper pairs, with negligible effect on the ground-state energy. The Hubbard model is a particularly nice example of the above phenomenon, which is revealed with the use of rigorous inequalities including the uncertainty principle inequality. In addition, we present numerical evidence that at half filling a related but simplified model shows off-diagonal long range order (ODLRO) of extended s-wave Cooper pairs.
In a Fermi system near or at half-filling, a specific superconducting pairing channel, if not explicitly included in the Hamiltonian, can be boosted by suppressing a competing pairing channel; this is exemplified by the enhancement of extended
s-wave correlations upon suppressing
s-wave Cooper pairing. This phenomenon, originally found by the use of generalized uncertainty relations is referred to as order by projection. The case of zero on-site Coulomb interaction in the thermodynamic limit, confirms this mechanism through the analytical solution. In this study, we go further and systematically investigate this mechanism for a strongly correlated fermionic Hubbard model, now with finite on-site interaction, on a square lattice with an extended set of hopping parameters. We explore the behaviors of different pairing channels when one of them is suppressed, utilizing density matrix renormalization group calculations. Our findings provide numerical evidence supporting the existence of order by projection in the strongly correlated system we studied. We also investigate the effect of the strength of Hubbard
U, next-nearest-neighbor ${t}^{\ensuremath{'}}$, hole doping, as well as finite-size scaling approaching the thermodynamic limit.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
We investigate the ionic Hubbard model (IHM) at half-filling in the limit of strong correlations and large ionic potential. The low-energy effective Hamiltonian in this limit, obtained by a similarity transformation, is a modified $t\ensuremath{-}J$ model with effective second-neighbor hopping terms. We explore the possibilities of
d-wave pairing and extended
s-wave pairing superconducting (SC) phases on a two-dimensional square lattice at zero temperature within a Gutzwiller projected renormalized mean-field theory. In the sector of solutions that forbid spin-ordering, the system shows a finite nonzero
d-wave as well as extended
s-wave pairing amplitude for $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U\ensuremath{\gg}t$. The width of the superconducting phase in $U\ensuremath{-}\mathrm{\ensuremath{\Delta}}$ regime shrinks with increase in
U and $\mathrm{\ensuremath{\Delta}}$, though the extended
s-wave pairing phase is higher in energy than the
d-wave pairing superconducting phase. But in a spin-resolved renormalized mean-field calculation, which allows for an antiferromagnetic (AF) order along with the
d-wave or extended
s-wave pairing, the SC phase is no longer viable and the system shows a direct transition from an AF ordered phase to a paramagnetic band insulator. Except for a thin sliver of a half-metallic AF phase close to the AF transition point, most of the AF ordered phase is a Mott insulator. We benchmarked the AF Mott insulator to band insulator transition within the Gutzwiller projected renormalized mean-field theory against the dynamical mean-field theory solved using continuous time quantum Monte Carlo. Our work suggests that the ground-state phase diagram of the IHM at half-filling in the limit of extreme correlations does not have any SC phase. The SC phase seen in the paramagnetic sector is a metastable phase, being higher in energy than the AF Mott insulator phase.
The two-dimensional Hubbard model near half-filling is studied using a configuration-interaction method. Superconducting order-parameter and spin-spin correlation functions are calculated as functions of distance to probe for superconductivity and antiferromagnetism. Consistent with our previous results, these more extensive calculations point toward d-wave superconductivity in the ground state of the two-dimensional repulsive Hubbard model. The d-wave correlations tend to increase with interaction strength, size of the state space used in the calculation, and doping away from half-filling. The superconducting correlations on the Fermi surface are interpreted as d-wave pairing between spin density-wave quasiparticles.
A theoretical study of BCS pairing in an extended Hubbard model with on-site repulsion (U) and a BCS pairing field (V) is presented. Treating the effect of U in the Gutzwiller approximation, we study the effect of increasing U in the Fermi-liquid phase, on the BCS pairing due to V. It is found that the superconducting energy gap (\ensuremath{\Delta}) is strongly enhanced in the correlated metallic phase near half-filling of the band due to the localization effects of U. Further, the ground state is found to be superconducting even when \ensuremath{\Vert}V\ensuremath{\Vert}U, contrary to the prediction in a Hartree-Fock treatment.