Quantum percolation transition in three dimensions: Density of states, finite-size scaling, and multifractality

The phase diagram of the metal-insulator transition in a three-dimensional quantum percolation problem is investigated numerically based on the multifractal analysis of the eigenstates. The large-scale numerical simulation has been performed on systems with linear sizes up to $L=140$. The multifractal dimensions, exponents $D_q$ and ${\alpha }_q$, have been determined in the range of $0\le q\le 1$. Our results confirm that this problem belongs to the same universality class as the three-dimensional Anderson model; the critical exponent of the localization length was found to be $\nu =1.622\pm 0.035$. However, the multifractal function $f\left(\alpha \right)$ and the exponents $D_q$ and ${\alpha }_q$ produced anomalous variations along the phase boundary, $p_c^Q\left(E\right)$.

Figure 1

Left side: Density of states of the quantum percolation model at different site-filling probabilities: (a) $p=0.35$, (b) $p=0.4$, (c) $p=0.5$, and (d) $p=0.6$. (e) Small clusters corresponding to special energies taken from the review of Schubert and Fehske [6].

Figure 2

Hopping elements in the “renormalized” Hamiltonian, ${\mathcal{H}}^2$.

Figure 3

First row: Eigenvectors of the Anderson model at $E=0$ (a) on the metallic side at $W=14$, (b) close to criticality $W=16.5$, and (c) on the insulating side at $W=20$. Second row: eigenvectors of the quantum percolation model at energy $E=0.1$, (d) on the metallic side at $p=0.5$, (e) close to criticality, $p=0.4535$, and (f) on the insulating side at $p=0.4$. Box sizes correspond to (a),(d) $400\times \sqrt{{\left|\Psi \right|}^2}$; (b),(e) $70\times \sqrt{{\left|\Psi \right|}^2}$; and (c),(f) $20\times \sqrt{{\left|\Psi \right|}^2}$. Multiplying factors were tuned to best sight but without overlapping cubes. System size, $L=120$, for all subfigures. Coloring is due to the $x$ coordinate.

Figure 4

The generalized multifractal exponents (a) ${\tilde{\alpha }}_1^{\text{ens}}(p,L,\ell =1)$ at $E=0.7$ and (b) ${\tilde{D}}_{0.5}^{\text{ens}}(p,L,\ell =1)$ at $E=0.1$ for the 3D quantum percolation model. Points with error bars are the raw data, and red solid lines are the best fits of the function Eq. (29) as a function of disorder, $p$, at different system sizes, $L$.

Figure 5

Critical point (left column) and critical exponent (right column) of the 3D quantum percolation model at (a) and (b) $E=0.1$, (c) and (d) $E=0.7$, and (e) and (f) $E=3.1$. Error bars represent $95\%$ confidence levels.

Figure 6

(a) Mobility edge for the 3D quantum percolation model; the dotted line denotes the classical percolation threshold, $p_c^C=0.3116\pm 0.0002$ [3]. Circles are approximate values of the bandwidth; beyond them only the Lifshitz tail is present. Squares are results from MFSS, and the line is a spline to guide the eye. Empty squares and dashed lines are for approximate data obtained from MFSS at $E=0.001$ and $0.01$. (b) Critical exponent for the 3D quantum percolation model. Error bars are for the $95\%$ confidence band. The dashed line is the average, and dotted lines note the $95\%$ confidence band around the average. The resulting critical exponent is $\nu =1.622(1.587,\cdots ,1.658)$.

Figure 7

Mobility edge of the 3D quantum percolation model in the literature [3, 6, 8, 9, 22].

Figure 8

First row: (a) $D_q$, (b) ${\alpha }_q$ as a function of $q$ at different energies, $E$. Second row: GMFEs shifted by their value for the Anderson model, (a) $D_q^{\text{QP}}-D_q^{\text{And}}$ and (b) ${\alpha }_q^{\text{QP}}-{\alpha }_q^{\text{And}}$, as a function of $p_c^Q$ at different $q$ values. Dotted lines are guides for the eye, error bars represent a $95\%$ confidence band on (c) and (d). Insets are the same, but without the shift.

Figure 9

Symmetry relation of ${\Delta }_q$ (left column) and ${\alpha }_q$ (right column) of the 3D quantum percolation model at (a) and (b) $E=0.1$, (c) and (d) $E=0.7$ (at the bottom of the mobility edge), and (e) and (f) $E=3.1$. Error bars are $95\%$ confidence levels. Points are naturally symmetric (antisymmetric) for $q=0.5$ for ${\alpha }_q$ (${\Delta }_q$) because of the addition (subtraction) of terms corresponding to $q$ and $1-q$.

Figure 10

$f\left(\alpha \right)$ obtained from $D_q$ and ${\alpha }_q$ computed at different energies, meaning different $p_c^Q$. Pentagons are the results for the Anderson model, the solid line is a fourth-order polynomial, and the dashed line is a second-order polynomial. Insets are magnified parts of the curve.