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Physical review. B, Condensed matter

Real-space scaling methods applied to an interacting fermion Hamiltonian

G. Spronken, Rémi Jullien, M. Avignon

Published Nov 1, 1981• 48 citations
Publisher

Abstract

Two different real-space techniques, the renormalization-group blocking method and finitecell scaling, are applied to a Hamiltonian of interacting fermions in one dimension which considers repulsion between electrons on neighboring sites. The localization transition which occurs at $T=0$ when the strength of the electron repulsion increases is qualitatively well described by both techniques. However, the essential singularity which appears in the exact solution is only recovered by the finite-cell-scaling method.

DOI
10.1103/physrevb.24.5356

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Two different real-space techniques, the renormalization-group blocking method and finitecell scaling, are applied to a Hamiltonian of interacting fermions in one dimension which considers repulsion between electrons on neighboring sites. The localization transition which occurs at T=0T=0T=0 when the strength of the electron repulsion increases is qualitatively well described by both techniques. However, the essential singularity which appears in the exact solution is only recovered by the finite-cell-scaling method.
We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength WWW, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping VL{V}_{L}VL​ between two configurations separated by a distance LLL in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength Wc{W}_{c}Wc​. In the localized phase W>WcW>{W}_{c}W>Wc​, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in LLL as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase W<WcW<{W}_{c}W<Wc​, the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.
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.
Although originally thought to show great promise in solving quantum many-body problems on a lattice, numerical real-space renormalization-group techniques have had little success for such problems. We explore the nature of the difficulties involved by studying the applicaton of the method to the simple tight-binding model in one dimension. The standard approach fails dramatically for this model. We show that the key to successfully applying the renormalization-group technique lies in applying a variety of boundary conditions to a block in order to simulate the effect of neighboring blocks.
Sarker's real-space renormalization-group method, applied previously to the one-dimensional dimerized XY model, is used to study the one-dimensional pure and dimerized Heisenberg models. Comparison of critical exponents (the dynamical and the gap exponents) with exact or conjectured values indicates that the method, in its original version, does not yield an improved real-space technique, at least for the models studied for which the opening of the gap leads to essential singularity or logarithmic corrections.