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arXiv:0905.4127 · Physical Review B

Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

Cécile Monthus, Thomas Garel

Published Jul 16, 2009• 28 citations
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Abstract

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$, we study numerically the statistical properties of the renormalized on-site energies $ϵ$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ϵ$ remain finite in the localized phase in $d=2,3$ and at criticality $(d=3)$, with a finite density at $ϵ=0$ and a power-law decay $1/{ϵ}^{2}$ at large $|ϵ|$. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality $(d=3)$, the statistics of renormalized hoppings $V$ is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

DOI
10.1103/physrevb.80.024203

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We perform numerical calculations on a simple cubic lattice for a standard diagonally disordered tight-binding Hamiltonian, whose random site energies are chosen from a Gaussian distribution with variance ∑2. From phenomenological renormalization group studies of the localization length, we determine that the critical disorder is σc≡∑c/J=6.00±0.17, which is in good agreement with previous results (J is the nearest neighbor transfer matrix element). From our calculations we can also determine the mobility edge trajectory, which appears to be analytic at the band center. Defining an order parameter exponent β, which determines how the fraction of extended states vanishes as the critical point is approached from below, this implies that β=1/2, in agreement with a previous study. From a finite-size scaling analysis we find that π2/ν=1.43±0.10, where π2 and ν are the inverse participation ratio and localization length critical exponents, respectively. This ratio of exponents can also be interpreted as the fractal dimension (also called the correlation dimension) D2 of the critical wave functions. Generalizations of the inverse participation ratio lead to a whole set of critical exponents πk, and corresponding generalized fractal dimensions Dk=πk/ν(k−1). From finite-size scaling results we find that D3=1.08±0.10 and D4=0.87±0.09. The inequality of the three dimensions D2, D3, and D4 shows that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the participation ratios, we find that ν=0.99±0.04. This result is in agreement with experiments on compensated or amorphous doped semiconductors.
Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).
Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions DqD_qDq​. The RG flows recover the qualitative structure previously reported for the information dimension D1D_1D1​, while their extension to other DqD_qDq​ with smaller moment orders q<1/2q<1/2q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1D_1D1​ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2q>1/2q>1/2 and small q<1/2q<1/2q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.