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arXiv:0804.3872 · Physical Review B

Delocalization effect of the Hubbard repulsion in exact terms and two dimensions

Zsolt Gulácsi

Published Jun 11, 2008• 16 citations
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Abstract

The genuine physical reasons explaining the delocalization effect of the Hubbard repulsion $U$ are analyzed. First, it is shown that always when this effect is observed, $U$ acts on the background of a macroscopic degeneracy present in a multiband type of system. After this step, I demonstrate that acting in such conditions, by strongly diminishing the double occupancy, $U$ spreads out the contributions in the ground state wave function, hence strongly increases the one-particle localization length, and consequently extends the one-particle behavior producing conditions for a delocalization effect. To be valuable, the reported results are presented in exact terms, being based on the first exact ground states deduced at half-filling in two dimensions for a prototype two band system, which is the generic periodic Anderson model at finite value of the interaction.

DOI
10.1103/physrevb.77.245113

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The genuine physical reasons explaining the delocalization effect of the Hubbard repulsion UUU are analyzed. First, it is shown that always when this effect is observed, UUU acts on the background of a macroscopic degeneracy present in a multiband type of system. After this step, I demonstrate that acting in such conditions, by strongly diminishing the double occupancy, UUU spreads out the contributions in the ground state wave function, hence strongly increases the one-particle localization length, and consequently extends the one-particle behavior producing conditions for a delocalization effect. To be valuable, the reported results are presented in exact terms, being based on the first exact ground states deduced at half-filling in two dimensions for a prototype two band system, which is the generic periodic Anderson model at finite value of the interaction.
We study numerically the ground-state properties of the repulsive Hubbard model for spin-1/2 electrons on two-dimensional lattices with disordered on-site energies. The projector quantum Monte Carlo method is used to obtain very accurate values of the ground-state charge density distributions with Np{N}_{p}Np​ and Np+1{N}_{p}+1Np​+1 particles. The difference in these charge densities allows us to study the localization properties of an added particle. The results obtained at quarter filling on finite clusters show that the Hubbard repulsion has a strong delocalizing effect on the electrons in disordered two-dimensional lattices. However, numerical restrictions do not allow us to reach a definite conclusion about the existence of a metal-insulator transition in the thermodynamic limit in two dimensions.
We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.
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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 (ty{\mathit{t}}_{\mathit{y}}ty​=tx{\mathit{t}}_{\mathit{x}}tx​); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=ty{\mathit{t}}_{\mathit{y}}ty​/tx{\mathit{t}}_{\mathit{x}}tx​, is different from the unity and zero the ground state is a conductor up to a critical value (U/t)c1{)}_{\mathit{c}1})c1​, 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)c2{)}_{\mathit{c}2})c2​\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.
Using a combination of ground state quantum Monte Carlo and finite size scaling techniques, we perform a systematic study of the effect of Coulomb interaction on the localization length of a disordered two-dimensional electron gas. We find that correlations delocalize the 2D system. In the absence of valley degeneracy (as in GaAs heterostructures), this delocalization effect corresponds to a finite increase of the localization length. The delocalization is much more dramatic in the presence of valley degeneracy [as in Si metal-oxide-semiconductor field-effect transistors], where the localization length increases drastically. We find that a simple mechanism accounts for the main features of the metallic behavior observed in two-dimensional gases. Our findings support the claim that this behavior is a genuine effect of the presence of interactions, yet that the system is not a "true" metal in the thermodynamic sense.