Metal-insulator transition in a two-dimensional system of chiral unitary class

We perform a numerical investigation of Anderson metal-insulator transition (MIT) in a two-dimensional system of chiral symmetry class AIII by combining finite-size scaling, transport, density of states, and multifractality studies. The results are in agreement with the $\sigma$-model renormalization-group theory where MIT is driven by proliferation of vortices. We determine the phase diagram and find an apparent nonuniversality of several parameters on the critical line of MIT, which is consistent with the analytically predicted slow renormalization towards the ultimate fixed point of the MIT. The localization-length exponent $\nu$ is estimated as $\nu =1.55\pm 0.1$.

Figure 1

Schematic of the RG flow implied by Eqs. (1, 2, 3). The starting value $y_0$ of the fugacity is taken to be the critical one $y_0=\frac{1}{4}$ and the resulting flow is projected to the $\sigma -\kappa$ plane.

Figure 2

Finite-size scaling analysis. Ratio ${\xi }_M/M$ as a function of staggering $\delta$ for disorder $W=0.5$ and $M=12,...,256$. The inset: data collapse ${\xi }_M/M=F\left(d^{\nu }M\right)$ with $d=\delta -{\delta }_c$, critical staggering ${\delta }_c=1.22$, and the exponent $\nu =1.55$.

Figure 3

Phase diagram of MIT on the ($W, \delta$) plane. Red symbols: MIT critical values ${\delta }_c\left(W\right)$ obtained by transfer-matrix analysis, see Fig. 2. Color code: Inverse participation ratio (IPR) exponent, ${\tau }_2\left(L\right)=-\partial ln{P}_2\left(L\right)/\partial lnL$ for largest available $L$'s.

Figure 4

Exponent ${\alpha }_{\nu }(W,\delta )$ of the DOS scaling, $\nu \left(\epsilon \right)\propto {\epsilon }^{{\alpha }_{\nu }}$ across the MIT at $W=0.3,0.5,1,2,3$. Positions of the MIT critical points ${\delta }_c\left(W\right)$ and the corresponding values ${\alpha }_{\nu }(W,{\delta }_c\left(W\right))$ are marked by dots.

Figure 5

Small symbols: ratio ${\xi }_M/M$ from transfer-matrix analysis of a quasi-1D system with $W=0.3$, 0.5, 1, and 3 (cf. Fig. 2). Large symbols: $1/\left[2\pi ({\alpha }_0-2+x_{\nu })\right]$ obtained by multifractal analysis of a 2D system with $L\le 96$ (blue) and $L\le 768$ (red). The relation (8) is fulfilled in the metallic phase and at criticality, see also Table 1.