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Register a new userMultifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
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We study the multifractal analysis (MFA) of electronic wavefunctions at the localisation-delocalisation transition in the 3D Anderson model for very large system sizes up to $240^3$. The singularity spectrum $f(alpha)$ is numerically obtained using the textsl{ensemble average} of the scaling law for the generalized inverse participation ratios $P_q$, employing box-size and system-size scaling. The validity of a recently reported symmetry law [Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analysed at the metal-insulator transition (MIT). The results are compared to those obtained using different approaches, in particular the typical average of the scaling law. System-size scaling with ensemble average appears as the most adequate method to carry out the numerical MFA. Some conjectures about the true shape of $f(alpha)$ in the thermodynamic limit are also made.
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The multifractality of the critical eigenstate at the metal to insulator transition (MIT) in the three-dimensional Anderson model of localization is characterized by its associated singularity spectrum f(alpha). Recent works in 1D and 2D critical systems have suggested an exact symmetry relation in f(alpha). Here we show the validity of the symmetry at the Anderson MIT with high numerical accuracy and for very large system sizes. We discuss the necessary statistical analysis that supports this conclusion. We have obtained the f(alpha) from the box- and system-size scaling of the typical average of the generalized inverse participation ratios. We show that the best symmetry in f(alpha) for typical averaging is achieved by system-size scaling, following a strategy that emphasizes using larger system sizes even if this necessitates fewer disorder realizations.
We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wavefunction intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4times10^6$ independently generated wavefunctions of system sizes up to $L^3=150^3$, yields accurate estimates for the critical exponent of the localization length, $ u=1.446 (1.440,1.452)$, the critical value of the disorder strength and the multifractal exponents.
We propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wavefunction amplitudes is sufficient to characterize the transition. In combination with finite-size scaling, this formalism permits the critical parameters to be estimated without the need for conductance or other transport measurements. Applying this method to high-precision data for wavefunction statistics obtained by exact diagonalization of the three-dimensional Anderson model, we estimate the critical exponent $ u=1.58pm 0.03$.
We consider the orthogonality catastrophe at the Anderson Metal-Insulator transition (AMIT). The typical overlap $F$ between the ground state of a Fermi liquid and the one of the same system with an added potential impurity is found to decay at the AMIT exponentially with system size $L$ as $F sim exp (- langle I_Arangle /2)= exp(-c L^{eta})$, where $I_A$ is the so called Anderson integral, $eta $ is the power of multifractal intensity correlations and $langle ... rangle$ denotes the ensemble average. Thus, strong disorder typically increases the sensitivity of a system to an additional impurity exponentially. We recover on the metallic side of the transition Andersons result that fidelity $F$ decays with a power law $F sim L^{-q (E_F)}$ with system size $L$. This power increases as Fermi energy $E_F$ approaches mobility edge $E_M$ as $q (E_F) sim (frac{E_F-E_M}{E_M})^{- u eta},$ where $ u$ is the critical exponent of correlation length $xi_c$. On the insulating side of the transition $F$ is constant for system sizes exceeding localization length $xi$. While these results are obtained from the mean value of $I_A,$ giving the typical fidelity $F$, we find that $I_A$ is widely, log normally, distributed with a width diverging at the AMIT. As a consequence, the mean value of fidelity $F$ converges to one at the AMIT, in strong contrast to its typical value which converges to zero exponentially fast with system size $L$. This counterintuitive behavior is explained as a manifestation of multifractality at the AMIT.
The probability density function (PDF) for critical wavefunction amplitudes is studied in the three-dimensional Anderson model. We present a formal expression between the PDF and the multifractal spectrum f(alpha) in which the role of finite-size corrections is properly analyzed. We show the non-gaussian nature and the existence of a symmetry relation in the PDF. From the PDF, we extract information about f(alpha) at criticality such as the presence of negative fractal dimensions and we comment on the possible existence of termination points. A PDF-based multifractal analysis is hence shown to be a valid alternative to the standard approach based on the scaling of general inverse participation ratios.
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