Stability of d-wave superconductivity in the t-J model

We use a recently developed technique, which allows to perform few Lanczos steps 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 of the BCS d-wave variational wavefunction. Our statistical Lanczos algorithm, which extends and improves the one Lanczos step, has been extensively tested on the small L=18 sites cluster where many Lanczos iterations can be performed exactly. In this case, at doping $\delta\sim 10$ the BCS wavefunction represents a very good initial state to achieve extremely accurate energies and correlation 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 weakly affected 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 our calculation we have computed the variance of the Hamiltonian $\Delta E_p=(< \hat{H}^2 >-< \hat{H} >^2)/L^2$ on the BCS wavefunction with $p=0,1,2$ Lanczos steps. For large $p$, when the Lanczos algorithm converges to the exact ground state, the variance vanishes exponentially with increasing $p$. The remarkable reduction of the variance, observed for $p=1,2$ Lanczos steps even for the largest lattice size considered, suggests a smooth and rapid convergence to the exact ground state. These results support the existence of off-diagonal long-range d-wave superconducting order in the two-dimensional $t{-}J$ model.