A detailed study of the behavior of two electrons in the square lattice is presented in which the dynamics of the electronic pair is described by the t-J model. The conditions for the formation of bound states, pair correlation functions, and the energy spectra are obtained. The analysis is extended to the Cooper problem, i.e., to the study of the wave function of two electrons above the filled Fermi sea. Finally, the quantitative relationship between the t-J model and the one-band Hubbard model is numerically investigated for the two-electron problem.
We investigate the phase diagram for the $t\text{\ensuremath{-}}J$ model on a two-dimensional honeycomb structure at low electronic densities. The phase diagram as a function of the interaction strength,
J, consists of three phases, viz. a gas of electrons for $0\ensuremath{\leqslant}J<2t$, a gas of pairs for $2t\ensuremath{\leqslant}J<3.29t$, and a fully phase-separated state for
J>3.29t. Further, we rigorously prove that a gas of electrons start forming pairs at
J=2t and the value is the same for any bipartite lattice in any dimension. We have elaborately explored the possibilities of formation of bigger clusters, i.e., bound states of three, four, and six electrons by performing exact diagonalization and perturbative analysis and ruled out formation of such clusters in the phase diagram.
We analyze the interplay between superconductivity and the formation of bound pairs of fermions (BCS-BEC crossover) in a 2D model of interacting fermions with small Fermi energy
EF and weak attractive interaction, which extends to energies well above
EF. The 2D case is special because a two-particle bound state forms at arbitrary weak interaction, and already at weak coupling, one has to distinguish between the bound-state formation and superconductivity. We briefly review the situation in the one-band model and then consider two different two-band models: one with one hole band and one electron band and another with two hole or two electron bands. In each case, we obtain the bound-state energy
2E0 for two fermions in a vacuum and solve the set of coupled equations for the pairing gaps and the chemical potentials to obtain the onset temperature of the pairing
Tins and the quasiparticle dispersion at
T=0. We then compute the superfluid stiffness ${\ensuremath{\rho}}_{s}(T=0)$ and obtain the actual
Tc. For definiteness, we set
EF in one band to be near zero and consider different ratios of
E0 and
EF in the other band. We show that at ${E}_{F}\ensuremath{\gg}{E}_{0}$, the behavior of both two-band models is BCS-like in the sense that ${T}_{c}\ensuremath{\approx}{T}_{\mathrm{ins}}\ensuremath{\ll}{E}_{F}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{T}_{c}$. At ${E}_{F}\ensuremath{\ll}{E}_{0}$, the two models behave differently: in the model with two hole/two electron bands, ${T}_{\mathrm{ins}}\ensuremath{\sim}{E}_{0}/ln\frac{{E}_{0}}{{E}_{F}}, \mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{({E}_{0}{E}_{F})}^{1/2}$, and ${T}_{c}\ensuremath{\sim}{E}_{F}$, like in the one-band model. In between
Tins and
Tc, the system displays a preformed pair behavior. In the model with one hole and one electron bands,
Tc remains of order
Tins, and both remain finite at
EF=0 and of the order of
E0. The preformed pair behavior still does exist in this model because
Tc is numerically smaller than
Tins. For both models, we reexpress
Tins in terms of the fully renormalized two-particle scattering amplitude by extending to the two-band case (the method pioneered by Gorkov and Melik-Barkhudarov back in 1961). We apply our results for the model with a hole and an electron band to Fe pnictides and Fe chalcogenides in which a superconducting gap has been detected on the bands that do not cross the Fermi level, and to FeSe, in which the superconducting gap is comparable to the Fermi energy. We apply the results for the model with two electron bands to Nb-doped
SrTiO3 and argue that our theory explains the rapid increase of
Tc when both bands start crossing the Fermi level.