Variational Monte Carlo study on the superconductivity in the two-dimensional Hubbard model

Abstract

The possibility of superconductivity (SC) in the ground state of the two-dimensional (2D) Hubbard model was investigated by means of the variational Monte Carlo method. The energy gain of the d-wave SC state, obtained as the difference of the minimum energy with a finite gap and that with zero gap, was examined with respect to dependences on U, electron density ρ and next nearest neighbor transfer t′ mainly on the 10×10 lattice. It was found to be maximized around U=8 (the energy unit is nearest neighbor transfer t). It was shown to sharply increase for negative values of t′ and have a broad peak for t′∼−0.10. For these value of t′ the energy gain was a smooth increasing function of ρ almost independent of the shell structure in the region starting from ∼0.76 up to the upper bound of investigation 0.92. This clearly indicates that the result is already close to the value in the bulk limit. For t′=0, the energy gain depended on the electronic shell state. This suggests the 10×10 lattice is not sufficiently large for this case, although it is highly plausible that the bulk limit value is finite. Competition between the SC and the commensurate SDW states was also investigated. When t′=0, the ground state is SDW in the range of ρ≥∼0.84. The SC region slightly extends up to ∼0.87 for t′∼−0.10. Consequently the present results strongly support an assertion that the 2D Hubbard model with t′∼−0.1 drives SC by itself in the ρ region from ∼0.76 to ∼0.87. The above features are in a fair agreement with the phase diagram of the optimally and overly hole-doped cuprates. The energy gain in the SC state with suitable parameters is found to be in reasonable agreement with the condensation energy in the SC state of YBa 2Cu 3O 7. The corresponding t– J model proves to give an order-of-magnitude larger energy gain, which questions its validity.