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Physical review. B, Condensed matter

Superconductivity in correlated wave functions

Claudius Gros

Published Jul 1, 1988• 294 citations
Publisher

Abstract

We describe a new method to numerically evaluate the properties of correlated superconducting wave functions. We have applied it to the resonating-valence-bond (RVB) wave function for the Hubbard model on the square lattice. For the half-filled case we find that the $d$-wave RVB state and the antiferromagnetic ordered state have the same energy within numerical accuracy. At 10% doping we find $d$-wave superconductivity, consistent with previous studies. We show that the superconducting order parameter is proportional to the number of holes, for small hole concentrations.

DOI
10.1103/physrevb.38.931

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We describe a new method to numerically evaluate the properties of correlated superconducting wave functions. We have applied it to the resonating-valence-bond (RVB) wave function for the Hubbard model on the square lattice. For the half-filled case we find that the ddd-wave RVB state and the antiferromagnetic ordered state have the same energy within numerical accuracy. At 10% doping we find ddd-wave superconductivity, consistent with previous studies. We show that the superconducting order parameter is proportional to the number of holes, for small hole concentrations.
The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(l) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and we find that a d-wave RVB state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes.
The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(1) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and the authors find that a d-wave RVB state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes.
We evaluate numerically several superconducting correlation functions in a generalized t−Jt-Jt−J model derived for hole-doped CuO2_22​ planes. The model includes a three-site term t′′t''t′′ similar to that obtained in the large UUU limit of the Hubbard model but of opposite sign for realistic O-O hopping. For realistic parameters we obtain strong evidence of superconductivity of predominantly dx2−y2d_{x^2-y^2}dx2−y2​ character. The ground state has a large overlap with a very simple resonating-valence-bond wave function with off-diagonal long-range order. This function reproduces the main features of the magnetic and superconducting correlation functions.
We use an effective Hamiltonian for two-dimensional Hubbard model including an antiferromagnetic spin-spin coupling term to study recently proposed Gossamer superconductivity. We apply a renormalized mean field theory to approximately take into account the strong correlation effect in partially projected Gutzwiller wave functions. At the half-filling, there is a first order phase transition to separate a Mott insulator at large Coulomb repulsion UUU from a Gossamer superconductor at small UUU. At the critical value U=UcU={U}_{c}U=Uc​, the charge carrier density and the superconducting (SC) order parameter change discontinuously from zero in the Mott insulating phase to finite values in the Gossamer SC phase. The first order transition is due to the interplay of the kinetic and spin exchange energies. As the electron density changes away from half-filling, the Gossamer SC state changes smoothly, while the Mott insulator is evolved into a resonating valence bond (RVB) SC state. The Gossamer and RVB SC states have the same pairing symmetry. The SC order parameter changes smoothly from a RVB SC state at U>UcU>{U}_{c}U>Uc​ to a Gossamer SC state at U<UcU<{U}_{c}U<Uc​ at a fixed nonhalf-filled electron density. We argue that the RVB SC state is smoothly connected to the Gossamer SC state, hence to the BCS state.
The two-dimensional Hubbard model near half-filling is studied using a configuration-interaction method. Superconducting order-parameter and spin-spin correlation functions are calculated as functions of distance to probe for superconductivity and antiferromagnetism. Consistent with our previous results, these more extensive calculations point toward d-wave superconductivity in the ground state of the two-dimensional repulsive Hubbard model. The d-wave correlations tend to increase with interaction strength, size of the state space used in the calculation, and doping away from half-filling. The superconducting correlations on the Fermi surface are interpreted as d-wave pairing between spin density-wave quasiparticles.
In a recent review by Anderson and co-workers, it was pointed out that an early resonating valence bond (RVB) theory is able to explain a number of unusual properties of high-temperature superconducting (SC) Cu oxides. Here we extend previous calculations to study more systematically the low-energy physical properties of the plain vanilla ddd-wave RVB state, and to compare the results with the available experiments. We use a renormalized mean-field theory combined with variational Monte Carlo and power Lanczos methods to study the RVB state of an extended $t\text{\ensuremath{-}}J$ model in a square lattice with parameters suitable for the hole-doped Cu oxides. The physical observable quantities we study include the specific heat, the linear residual thermal conductivity, the in-plane magnetic penetration depth, the quasiparticle energy at the antinode $(\ensuremath{\pi},0)$, the superconducting energy gap, the quasiparticle spectra, and the Drude weights. The traits of nodes (including kF{k}_{F}kF​, the Fermi velocity vF{v}_{F}vF​, and the velocity along Fermi surface v2{v}_{2}v2​), and the SC order parameter are studied. Comparisons of the theory and the experiments in cuprates show an overall qualitative agreement, especially on their doping dependences.