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Physical review. B, Condensed matter

Ground-state energy of the one-dimensional Hubbard model with strong coupling

Glenn Fletcher

Published Jul 15, 1980• 0 citations
Publisher

Abstract

The ground-state energy has been calculated for a one-dimensional tight-binding Hubbard chain with nearest-neighbor hopping in the strong-coupling ($U=\ensuremath{\infty}$) limit using various approximate methods. The results are compared with the exact expression of Beni, Holstein, and Pincus to test the relative merits of different approximate schemes.

DOI
10.1103/physrevb.22.1096

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The ground-state energy has been calculated for a one-dimensional tight-binding Hubbard chain with nearest-neighbor hopping in the strong-coupling (U=\ensuremath∞U=\ensuremath{\infty}U=\ensuremath∞) limit using various approximate methods. The results are compared with the exact expression of Beni, Holstein, and Pincus to test the relative merits of different approximate schemes.
The dimerized one-dimensional Hubbard model is studied in the framework of lattice density-functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength U/tU/tU/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij{t}_{\mathrm{ij}}tij​ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
I apply a two-step density-matrix renormalization group method to the anisotropic two-dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as $24\ifmmode\times\else\texttimes\fi{}25$. I then apply the method to the interacting case. I find that for strong Hubbard interaction, the ground state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and is not enhanced by frustration. For weak Hubbard couplings, due to interchain interaction, interchain nonlocal singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. The pairing mechanism is reminiscent of the Kohn-Luttinger mechanism.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.