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

Abstract

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 p _d 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 $\Delta {E_p} = (\left\langle {{{\hat H}^2}} \right\rangle - {\left\langle {\hat H} \right\rangle ^2})/{L^2}$ 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.