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 [1]. Our statistical Lanczos algorithm, which extends and improves the one Lanczos step proposed in Ref. [2], has been extensively tested on the small L = 18 sites cluster where many Lanczos iterations can be performed exactly. In this case, at doping δ ∼ 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 ≤ 98) the behavior is similar: the low-energy d-wave order parameter Pd is weakly affected by a couple of Lanczos iterations in the low doping δ ∼ 10% region, whereas the energy is considerably lowered. As a further test of our calculation we have computed the variance of the Hamiltonian ∆Ep = (〈Ĥ 〉− 〈Ĥ〉)/L 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.