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Physical review. B, Condensed matter

Real-space renormalization-group study of the two-dimensional Hubbard model

Jesús Pérez-Conde, Pierre M. Pfeuty

Published Jan 1, 1993• 16 citations

Abstract

The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (${\mathit{t}}_{\mathit{y}}$=${\mathit{t}}_{\mathit{x}}$); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=${\mathit{t}}_{\mathit{y}}$/${\mathit{t}}_{\mathit{x}}$, is different from the unity and zero the ground state is a conductor up to a critical value (U/t${)}_{\mathit{c}1}$, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t${)}_{\mathit{c}2}$\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.

DOI: 10.1103/physrevb.47.856

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The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (

{\mathit{t}}_{\mathit{y}}

=

{\mathit{t}}_{\mathit{x}}

); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=

{\mathit{t}}_{\mathit{y}}

/

{\mathit{t}}_{\mathit{x}}

, is different from the unity and zero the ground state is a conductor up to a critical value (U/t

{)}_{\mathit{c}1}

, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t

{)}_{\mathit{c}2}

\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.

The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.

We investigate the application of the density matrix renormalization group (DMRG) to the Hubbard model in momentum space. We treat the one-dimensional models with dispersion relations corresponding to nearest-neighbor hopping and

1/r

hopping and the two-dimensional model with isotropic nearest-neighbor hopping. By comparing with the exact solutions for both one-dimensional models and with exact diagonalization in two dimensions, we first investigate the convergence of the ground-state energy. We find variational convergence of the energy with the number of states kept for all models and parameter sets. In contrast to the real-space algorithm, the accuracy becomes rapidly worse with increasing interaction and is not significantly better at half filling. We compare the results for different dispersion relations at fixed interaction strength over bandwidth and find that extending the range of the hopping in one dimension has little effect, but that changing the dimensionality from one to two leads to lower accuracy at weak-to-moderate interaction strength. In the one-dimensional models at half filling, we also investigate the behavior of the single-particle gap, the dispersion of spinon excitations, and the momentum distribution function. For the single-particle gap, we find that proper extrapolation in the number of states kept is important. For the spinon dispersion, we find that good agreement with the exact forms can be achieved at weak coupling if the large momentum-dependent finite-size effects are taken into account for nearest-neighbor hopping. For the momentum distribution, we compare with various weak-coupling and strong-coupling approximations and discuss the importance of finite-size effects as well as the accuracy of the DMRG.

The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.