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Physical review. B, Condensed matter

Analytical wave function and reduced Hamiltonian for the weakly coupled two-dimensional Hubbard model

Greg Levine, W. P. Su, Barry Friedman

Published Oct 1, 1992• 3 citations
Publisher

Abstract

We introduce a many-body wave function that is shown to describe accurately the ground state of the positive-U, two-dimensional Hubbard model on finite lattices close to half-filling. The wave function is used to demonstrate the absence of a conventional BCS ground state and pairing instability. A simple reduced Hamiltonian describing transverse spin fluctuations is obtained as the fixed-point Hamiltonian in the limit U\ensuremath{\simeq}\ensuremath{\Delta}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{s}}$ where \ensuremath{\Delta}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{s}}$ is the single-particle energy-level spacing at the Fermi surface.

DOI
10.1103/physrevb.46.8421

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We introduce a many-body wave function that is shown to describe accurately the ground state of the positive-U, two-dimensional Hubbard model on finite lattices close to half-filling. The wave function is used to demonstrate the absence of a conventional BCS ground state and pairing instability. A simple reduced Hamiltonian describing transverse spin fluctuations is obtained as the fixed-point Hamiltonian in the limit U\ensuremath{\simeq}\ensuremath{\Delta}\ensuremathϵs{\mathrm{\ensuremath{\epsilon}}}_{\mathit{s}}\ensuremathϵs​ where \ensuremath{\Delta}\ensuremathϵs{\mathrm{\ensuremath{\epsilon}}}_{\mathit{s}}\ensuremathϵs​ is the single-particle energy-level spacing at the Fermi surface.
We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
We show that for low and moderate fillings the ground state of the two-dimensional Hubbard model with positive-U and nearest-neighbor hopping is unstable with respect to d-wave pairing with dxy{\mathit{d}}_{\mathit{x}\mathit{y}}dxy​ symmetry of the gap wave function: \ensuremath{\Delta}\ensuremath{\sim}sinkx{\mathit{k}}_{\mathit{x}}kx​sinky{\mathit{k}}_{\mathit{y}}ky​. The inclusion of the next-neighbor hopping may either suppress the pairing or change the symmetry of the superconducting 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.