Real Space Numerical Renormalization Groups for Hubbard Models

Jesús Pérez-Conde

Osamu Sakai, Yoshio Kuramoto

Donovan James McMurrich Allum

S. Johri and R. Bhatt developed a real-space renormalization group approach aimed at extracting the localized single-particle eigenstates of the Anderson model from a large system by identifying clusters of resonant site potentials. E. Campbell generalized this real-space renormalization group approach using standard perturbation theory. Both approaches were intended to approximate the single-particle density of states of the Anderson model. In this thesis, we aimed to test the potential of applying a similar real-space renormalization group approach to calculate the density of states of the interacting Anderson-Hubbard model. Our interest in the density of states of this model is due to a V-shaped zero-bias anomaly in two-dimensional systems. A real-space renormalization group approach is best applied to a one-dimensional system. We found that the zero-bias anomaly is not V-shaped in one-dimension. To test the potential of a real-space renormalization group approach, we used the cluster approach which is the same as the non-interacting renormalization group approach but without the perturbation theory and found that for strong disorder this technique could accurately calculate the density of states over a wide range of energies but deviated from exact results at the band edge, at $omega=pm U$ and near $omega=0$. The first two inaccuracies will be reduced with a proper real-space renormalization group approach. We suspect that the last inaccuracy is associated with long range physics and may be difficult to recover. We also developed a technique that adjusts the identification of clusters in the cluster approach to improve the computation time of the density of states with minimal loss of accuracy in a tunable range around the Fermi level. We found that this technique significantly reduced the computation time and was able to preserve the density of states near the Fermi level, except at the smallest energies near $omega=0$. Author Keywords: Anderson-Hubbard model, renormalization group, Strong electron correlations, Zero-bias anomaly

Sven Binder

Michael Ma

We study the half-filled Hubbard model with random on-site energies with the use of a real-space renormalization group. For $d=1$, we find a transition from an Anderson insulator to a Hubbard insulator as the Hubbard $U$ is increased. We find no metallic phase. For $d=3$, we find a metallic phase which, surprisingly, is stabilized against Anderson localization by a small Hubbard $U$.

B. Fourcade, G. Spronken

The real-space renormalization-group method is applied to the one-dimensional extended Hubbard model. It is shown that in the half-filled-band case, the phase diagram consists of two phases and that the transition from the spin-ordered phase to the charge-ordered phase is continuous, a result which differs from that obtained previously from the broken-symmetry Hartree-Fock approximation.