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Physical review. B, Condensed matter

Hole-hole effective interaction in the two-dimensional Hubbard model

G. Fano, F. Ortolani, Alberto Parola

Published Oct 1, 1990• 79 citations
Publisher

Abstract

We present the first exact results of the ground-state energy in the two-dimensional Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster for several values of U and fillings. The behavior of the binding energy of two holes as a function of U is discussed together with some analytical results in limiting cases. The analysis of the hole-hole correlation function allows one to determine the range of values of U where binding might be seen.

DOI
10.1103/physrevb.42.6877

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We present the first exact results of the ground-state energy in the two-dimensional Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster for several values of U and fillings. The behavior of the binding energy of two holes as a function of U is discussed together with some analytical results in limiting cases. The analysis of the hole-hole correlation function allows one to determine the range of values of U where binding might be seen.
The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U)0.70{)}^{0.70})0.70, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole m∗{\mathit{m}}^{\mathrm{*}}m∗=8t/W (W represents the bandwidth) is proportional to U/T (m∗{\mathit{m}}^{\mathrm{*}}m∗\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value m∗{\mathit{m}}^{\mathrm{*}}m∗\ensuremath{\approxeq}84 for large U/t.
Using the Bethe ansatz, we calculate the binding energies of two holes just off half-filling in one-dimensional Hubbard chains. Somewhat surprisingly, we find binding comparable to that seen in two-dimensional systems in chains that have a degeneracy at the Fermi level in the noninteracting (U=0) limit. The binding initially IincreasesR with lattice size before beginning a slow decay to zero in the thermodynamic limit. We discuss the general implications of our results for binding energy and other numerical calculations of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ models.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
The authors have analysed finite clusters with different degrees of frustration on the basis of the conventional Hubbard model. In particular they have calculated the ground-state energy and certain correlation functions as a function of the interaction energy U. As U varies, a sharp transition is observed between two different structures for half-filled bands. The authors have studied these clusters not only for the case of a half-filled band but also for different numbers of holes. They found that the structure of the ground state depends critically on the total spin and the number of holes.