We investigate the one-dimensional t - J - J ′ model by the numerical diagonalization where a few holes are doped in the half-filled band. By introducing a finite-size scaling, we estimate the spin gap at J ′ / J =0.5 and J / t =2. When one hole is doped, the spin gap decreases to 0.190 J in comparison with 0.236 J which is the spin gap of the half-filled band. The spin gap further reduces to 0.141 J for two-hole doping and to about 0.05 J for three-hole doping. This result shows that the value of the spin gap as a function of the electron density is discontinuous at half-filling.
The one-dimensional t-t'-J-J' model is examined by numerical diagonalization in the case that one hole is doped to the half-filled band. In the presence of t', two types of solitons are found; one is a bound object of the hole and a singlet electron pair in a ferromagnetic background (t'>0, J0), and the other is that of the hole and a triplet electron pair in a singlet background (t'0, J>0). It seems that the solitons remain in the ground state when the system size becomes infinite. The formation of the solitons is ascribed to a kinematic exchange effect due to the t' term.
We consider an exactly solvable two-band model of electrons moving in one dimension and interacting with a $\ensuremath{\delta}$-function spin-exchange potential. The relative population of the bands is determined by the band splitting, the magnetic field, and the temperature. The interaction leads to the formation of spin-triplet bound states of the Cooper type (hard-core bosons that do not condensate and exist at all temperatures). In the zero-field ground state the spin-triplet states are effectively free bosons. A threshold band splitting is required to break up a triplet pair. The low-temperature and finite size properties are discussed in the context of anomalous superconducting and antiferromagnetic fluctuations.
The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J
Jc\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}
Jc the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.
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.
The one-dimensional Hubbard model with first- and second-neighbor hopping, t1 and t2, with t2≊t1/(2)1/2 and large repulsion U, is proposed as a testing ground of Anderson’s idea that the existence of an resonating valence bond ground state in the half-filled-band case will show superconductivity when a finite concentration of holes is introduced. Interestingly, our preliminary calculations in the U=∞ limit for small chains have revealed a new mechanism for enhancement of singlet pairing: We find that for t2<0 the ground state is a singlet, and, for t2<0 and ‖t2/t1‖≪1, the nearest-neighbor extended-singlet-pairing correlation function becomes enhanced appreciably as the temperature decreases below t2.