Inverse Anderson Transition Caused by Flatbands

We propose a new disorder-induced insulator-metal transition of one-electron states, which may be called the “inverse Anderson transition.” We first make a highly degenerated localized states by constructing a three-dimensional periodic system possessing only flat dispersion relations. When we introduce a disorder into it, a finite-size scaling of the level statistics shows two clear (localization-delocalization and delocalization-localization) transitions for a wide range of the energy, with increasing the degree of disorder. These transitions are confirmed also by finding the system-size-independent $f(\alpha )$ characteristic of the wave function.

Figure 1

Diamond lattice with fourfold-degenerated orbitals on each site. We consider the transfer integrals between orbitals within the nearest-neighbor sites. The two atoms $A$ and $B$ and eight orbits form a unit cell.

Figure 2

Dispersion relations of the flatband model. The matrix elements of the Hamiltonian are selected as $t_{A1,A2}=0.0$, $t_{A1,B1}=-1.0$, $t_{A1,B2}=1.0$, $t_{A2,B4}=-1.0$, $t_{A2,B3}=-1.0$, as in Ref. 28, so that the entire dispersion relations become flat and fourfold degenerated. Cf. the indices of orbitals in Fig. 1.

Figure 3

Nearest-neighbor level-spacing distribution function $P(s)$ for different degrees of disorder; $W=12$, $W=26$, and $W=50$, at $E=0$ for the case of $N=7$. Wigner (solid line) and Poisson (dotted line) functions are also shown for comparison.

Figure 4

The parameter $\gamma$ as a function of $W$ at $E=0$ for different system sizes. We can find two intersecting points at which the system-size-invariant $P(s)$ appears.

Figure 5

The mobility-edge trajectory of the flatband model derived by the system-size-independent level statistics. There are two transition points over the entire range of energy. The mesh is the energy area in which we cannot estimate two transition points accurately. The marks ○ show the points on which the system-size-independent $f(\alpha )$ characteristics are confirmed.

Figure 6

The $f(\alpha )$ characteristics of the wave functions for $W=14.0$ and $W=37.0$ at $E=0$. Error bars are made by using 100 samples and are indicated on $f$ and $\alpha$ for $q=-5,-2,-1,0,1,2,5$ for the case $N=6,8,10$.