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We use a recently developed technique, which allows to perform few Lanczossteps on a given wavefunction even for large system sizes, to investigate the$t{-}J$ model in the physical parameter region and to check the stability ofthe BCS d-wave variational wavefunction. Our statistical Lanczos algorithm,which extends and improves the one Lanczos step, has been extensively tested onthe small L=18 sites cluster where many Lanczos iterations can be performedexactly. In this case, at doping $\delta\sim 10%$ the BCS wavefunctionrepresents a very good initial state to achieve extremely accurate energies andcorrelation functions with few Lanczos iterations. For large sizes ($L\le 98$)the behavior is similar: the low-energy d-wave order parameter $P_d$ is weaklyaffected by a couple of Lanczos iterations in the low doping $\delta\sim 10%$region, whereas the energy is considerably lowered. As a further test of ourcalculation we have computed the variance of the Hamiltonian $\Delta E_p=(<\hat{H}^2 >-< \hat{H} >^2)/L^2$ on the BCS wavefunction with $p=0,1,2$ Lanczossteps. For large $p$, when the Lanczos algorithm converges to the exact groundstate, the variance vanishes exponentially with increasing $p$. The remarkablereduction of the variance, observed for $p=1,2$ Lanczos steps even for thelargest lattice size considered, suggests a smooth and rapid convergence to theexact ground state. These results support the existence of off-diagonallong-range d-wave superconducting order in the two-dimensional $t{-}J$ model.

Stability of D-Wave Superconductivity in the t- J Model