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Physical review. B, Condensed matter

Binding of holes in the Hubbard model

Elbio R. Dagotto, Adriana Moreo, Robert L. Sugar, D. Toussaint

Published Jan 1, 1990• 74 citations
Publisher

Abstract

We present a new numerical method for the study of binding energies of particles in fermionic systems. Working with an imaginary chemical potential we can obtain results in the canonical ensemble by simple modifications of standard numerical techniques. We applied the technique to the two-dimensional Hubbard model observing binding of holes at half-filling on lattices of 4\ifmmode\times\else\texttimes\fi{}4 sites. For U/t=4 we estimate that the binding energy is \ensuremath{\Delta}=-0.10\ifmmode\pm\else\textpm\fi{}0.02.

DOI
10.1103/physrevb.41.811

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Physical review. B, Condensed matter•Jul 1, 1989•947 citations
We present a new numerical method for the study of binding energies of particles in fermionic systems. Working with an imaginary chemical potential we can obtain results in the canonical ensemble by simple modifications of standard numerical techniques. We applied the technique to the two-dimensional Hubbard model observing binding of holes at half-filling on lattices of 4\ifmmode\times\else\texttimes\fi{}4 sites. For U/t=4 we estimate that the binding energy is \ensuremath{\Delta}=-0.10\ifmmode\pm\else\textpm\fi{}0.02.
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.
We present a technique for a numerical analysis of the phase structure of the two-dimensional Hubbard model as a function of the hole chemical potential. The grand-canonical partition function for the model is obtained via Monte Carlo simulations. The dependence of the hole occupation number on the chemical potential and the temperature is evaluated. These calculations, together with a study of the Yang-Lee zeros of the grand-canonical partition function, show evidence of a phase transition at zero temperature and particle density below half filling. The binding energy of a pair of holes is calculated in the low-temperature regime and the possibility for pairing is explored.
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.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
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.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.