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Physical review. B, Condensed matter

Renormalization group and exact diagonalization studies of Hubbard rings with impurities

弘之 森, Minoru HAMADA

Published Feb 15, 1996• 13 citations
Publisher

Abstract

We studied mesoscopic Hubbard rings with impurities using the renormalization group (RG) technique and the exact diagonalizations. The exact diagonalization calculations showed that the charge stiffness normalized by the value of a clean system has a peak as a function of electron-electron interactions. Previous RG analysis has succeeded in showing the enhancement of the stiffness at weak interactions, but does not reveal the peak behavior. We derived RG equations with a 4${\mathit{k}}_{\mathit{F}}$ impurity scattering term, which was ignored in the previous studies, and reproduced the overall behavior of the exact diagonalization results. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.4850

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We studied mesoscopic Hubbard rings with impurities using the renormalization group (RG) technique and the exact diagonalizations. The exact diagonalization calculations showed that the charge stiffness normalized by the value of a clean system has a peak as a function of electron-electron interactions. Previous RG analysis has succeeded in showing the enhancement of the stiffness at weak interactions, but does not reveal the peak behavior. We derived RG equations with a 4kF{\mathit{k}}_{\mathit{F}}kF​ impurity scattering term, which was ignored in the previous studies, and reproduced the overall behavior of the exact diagonalization results. \textcopyright{} 1996 The American Physical Society.
We studied mesoscopic Hubbard rings with impurities using the renormalization group (RG) technique and the exact diagonalizations. The exact diagonalization calculations showed that the charge stiffness normalized by the value of clean system has a peak as a function of electron-electron interactions. Previous RG analysis has succeeded to show the enhancement of the stiffness at weak interactions, but does not show the peak behavior. We derived RG equations with 4kF4k_F4kF​ impurity scattering term, which was ignored in the previous studies, and reproduced the overall behavior of the exact diagonalization results.
The stability of the 1D Luttinger liquid phase of interacting spinless particles (t - V model) to an on-site diagonal disorder is studied by exact diagonalization of small 1D rings up to 22 sites. We discuss our numerical results in the light of the phase diagram predicted by the Renormalization Group analysis. In a finite range of attractive interactions (around V/t ∼-0.5) the metallic phase (with superconducting fluctuations) seems to be stable up to some small value of the disorder parameter. On the other hand, in the repulsive case, the impurity potential clearly leads to a strong localization of the particles.
We investigate, by numerically calculating the charge stiffness, the effects of random diagonal disorder and electron-electron interaction on the nature of the ground state in the 2D Hubbard model through the finite-size exact diagonalization technique. By comparing with the corresponding 1D Hubbard model results and by using heuristic arguments we conclude that it is unlikely that there is a 2D metal-insulator quantum phase transition, although the effect of interaction in some range of parameters is to substantially enhance the noninteracting charge stiffness.
The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than 1/N.1/N.1/N. Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.
The persistent current and charge stiffness of a one-dimensional Luttinger liquid on a ring threaded by a magnetic flux are calculated by Monte Carlo simulation. By changing the random impurity potential strength and the electron-electron interaction, we see a crossover behavior between weak- and strong-impurity limits. For weak-impurity potentials, interactions enhance impurity effects, that is, interactions decrease the current and the stiffness. On the other hand, interactions tend to screen impurities when the impurity potential is strong. Temperature dependence of the persistent current and the charge stiffness shows a peak at a characteristic temperature, consistent with a recent single-impurity study.
Many of the most interesting electronic behaviours currently being studied are associated with strong correlations. In addition, many of these materials are disordered either intrinsically or due to doping. Solving interacting systems exactly is extremely computationally expensive, and approximate techniques developed for strongly correlated systems are not easily adapted to include disorder. As a non-interacting disordered model, it makes sense to consider the Anderson model as a first step in developing an approximate method of solution to the interacting and disordered Anderson-Hubbard model. Our renormalization group (RG) approach is modeled on that proposed by Johri and Bhatt [23]. We found an error in their work which we have corrected in our procedure. After testing the execution of the RG, we benchmarked the density of states and inverse participation ratio results against exact diagonalization. Our approach is significantly faster than exact diagonalization and is most accurate in the limit of strong disorder. Author Keywords: disorder, localization, real-space renormalization, strong correlations