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arXiv:cond-mat/9904310

The Ground and Low Lying Excited States of the Two-Dimensional Hubbard Model

Yoshihiro Asai

Published Apr 22, 1999• 0 citations
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Abstract

We have studied the ground state of the two-dimensional (2D) Hubbard model by using a quantum monte method paying special attention to the shell structure effect on finite size clusters. Our calculations show there is a gap for spin excitations in the ground state and incommensurate peaks at $(π\pm δ, π)$ and $(π, π\pm δ)$ in the spin correlation function for a low lying excited state. In the ground state, the long range part of the d-wave superconducting correlation function is enhanced and the momentum distribution function at the Fermi level $(π,0)$ is rounded. The gap in spin excitations and the momentum distribution function rounding is consistent with opening a d-wave superconducting gap in the ground state.

DOI
10.48550/arxiv.cond-mat/9904310

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We have studied the ground state of the two-dimensional (2D) Hubbard model by using a quantum monte method paying special attention to the shell structure effect on finite size clusters. Our calculations show there is a gap for spin excitations in the ground state and incommensurate peaks at (π±δ,π)(π\pm δ, π)(π±δ,π) and (π,π±δ)(π, π\pm δ)(π,π±δ) in the spin correlation function for a low lying excited state. In the ground state, the long range part of the d-wave superconducting correlation function is enhanced and the momentum distribution function at the Fermi level (π,0)(π,0)(π,0) is rounded. The gap in spin excitations and the momentum distribution function rounding is consistent with opening a d-wave superconducting gap in the ground state.
The quasiparticle and spin excitation spectra in the 2D Hubbard model are calculated self-consistently as functions of momentum on the real frequency axis. Near half filling we obtain in the normal state a pseudogap in the density of states. The pairing state has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry. Below Tc{T}_{c}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The scattering rate and spin-lattice relaxation rate drop abruptly below Tc{T}_{c}Tc​. Our results are consistent with experiments on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.
Numerical results obtained from Monte Carlo simulation in the ground state and at finite temperatures as well as the exact diagonalization and the transfer matrix method are reported. Efficiency of the Monte Carlo method in the ground state is examined. Spin, charge and superconducting correlations are investigated for the Hubbard and the t - J model in one and two dimensions. The momentum distribution in the one-dimensional Hubbard model shows fermi-liquid-like behavior at least in two orders of magnitude smaller energy scale than the band width. The short-range incommensurate spin correlation is observed and analyzed in the doped Hubbard and the t - J model both in one and two dimensions. In the ground state, the superconducting correlation shows the absence of system size dependence in two dimensions at the filling smaller than 0.8 and in one dimension at any filling. The binding energy of two fermions and the spin and charge distortion around an itinerant fermion are also discussed in the t - J and the coupled spin-fermion model.
In the first part of this paper we explore a configuration-interaction approach to the two-dimensional Hubbard model near half filling. To reproduce the result of degenerate second-order perturbation theory it is necessary to include Fock states having kinetic energy within 4t of the lowest kinetic-energy states. Using the ground state from the largest configuration-interaction calculation, we calculate the extended s- and d-wave superconducting correlation functions as a function of distance. There are indications of a nondecaying long-distance tail in the d-wave correlation function, that is, d-wave superconductivity in the ground state of the repulsive Hubbard model. In the second part of the paper, we reexamine our effective Hamiltonian for the repulsive Hubbard model near half filling. Simple considerations lead us to a more intuitively plausible form consistent with our previous results and spin-rotational invariance.
We investigate the ground-state properties of the two-dimensional Hubbard model, based on the off-diagonal wave function variational Monte Carlo method. We use an optimized wave function that is improved from an initial one-body wave function by multiplying by multiple correlation operators that are given by the form of exp(−S-S−S)-type where SSS is a suitable operator. We examine the inhomogeneous ground state at near 1/8 doping in the strongly correlated region where the on-site Coulomb interaction is larger than the bandwidth. The ddd-wave superconductivity with a spatially oscillating gap function can coexist with the charge ordering in the ground state without magnetic ordering, and superconducting condensation energy increases by this coexistence. We also show the ddd-wave pair correlation function as a function of lattice sites. The correlation function indicates that the long-range superconducting order indeed exists, and also that the strong electron correlation suppresses the pair correlation function.
I present a scheme for performing weak-coupling expansions for the Hubbard model on square clusters near half-filling. Using this scheme the 6 × 6 cluster with 2, 3 and 4 holes is considered. I have calculated the ground-state quantum numbers for the 3- and 4-hole cases. In the 3-hole case I find a nonzero momentum near the (±π/2, ±π/2) points, while for the four-hole case the momentum is (0, 0). The calculation of low-lying excited states indicates the opening of a spin gap as one dopes away from half-filling.
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.