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

Abstract

Ž .Ž . 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 r and next nearest neighbor transfer t X mainly on the 10 = 10 lattice. It was found to be maximized around Ž .X U s 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 X ; y0.10. For these value of t X the energy gain was a smooth increasing function of r 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 X s 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 X s 0, the ground state is SDW in the range of r G; 0.84. The SC region slightly extends up to ; 0.87 for t X ; y0.10. Consequently the present results strongly support an assertion that the 2D Hubbard model with t X ; y0.1 drives SC by itself in the r 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 Cu O . The corresponding t-J model proves to give an 2 3 7order-of-magnitude larger energy gain, which questions its validity. q 1998 Elsevier Science B.V. All rights reserved.

t′- and t″-dependence of the bulk-limit superconducting condensation energy of the 2D Hubbard model

“…Condensation energy E cond for the SC state is obtained as the decrease per site of this E g from the normal state energy similarly calculated starting from the normal state trial wave function [6]. E cond for the commensurate SDW state is obtained similarly.…”

Superconductivity in a Hubbard-Fröhlich model and in cuprates

“…The idea behind this originally proposed by Anderson 4 is that mobile hole pairs are created via a strong on-site repulsion U. Results by Paramekanti et al 5 and Yamaji et al, 6 using a variational Monte Carlo ͑VMC͒ simulation with a ͑projected͒ BCS-type trial-wave function, appeared to back this up. More recently Yamaji et al 7 found a condensation energy in a Hubbard model with next-and third-neighbor hopping but not in the simplest nearest-neighbor Hubbard model.…”

“…6 Our variational parameters are g and the chemical potential that enters through the kinetic energy k ϵ −2t͑cos k x + cos k y ͒ − via u k…”

“…Two cases are investigated: ͑a͒ an on-site repulsion Ũ = 8 is used for comparison with results of Ref. 6 and ͑b͒ an infinite Ũ is used to see if the attraction induced by EPI, alone, is enough for the high-temperature superconductivity. We examine the d-wave ⌬ k = ⌬͑cos k x − cos k y ͒, extended s-wave ⌬ k = ⌬͑cos k x + cos k y ͒, and s-wave ⌬ k = ⌬ order parameters.…”

“…We examine the d-wave ⌬ k = ⌬͑cos k x − cos k y ͒, extended s-wave ⌬ k = ⌬͑cos k x + cos k y ͒, and s-wave ⌬ k = ⌬ order parameters. Following previous work, 6 to avoid zeros in the wave function, periodic boundary conditions are used in one direction and antiperiodic boundary conditions are used in the other. The antiperiodic boundary conditions lead to a phase factor that is accounted for in the algorithm.…”