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Physical review. B, Condensed matter

Hubbard model with one hole: Ground-state properties

Elbio R. Dagotto, Adriana Moreo, Ted Barnes

Published Oct 1, 1989• 79 citations
Publisher

Abstract

The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J${J}_{c}$\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}${J}_{c}$ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.

DOI
10.1103/physrevb.40.6721

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Physical review. B, Condensed matter•Oct 1, 1991•28 citations
The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling JJc{J}_{c}Jc​\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}Jc{J}_{c}Jc​ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.
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 exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.
The effective Hamiltonian, obtained from the Hubbard model in the strong-coupling limit, is diagonalized exactly for a periodic two-dimensional square lattice of ten sites. We obtain the ground state of the system for any filling of the lattice and the KKK-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio tU\frac{t}{U}Ut​, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing tU\frac{t}{U}Ut​, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of tU\frac{t}{U}Ut​ exists for which pairing of the holes may be possible.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.
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.