Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class

We study the multifractality (MF) of critical wave functions at boundaries and corners at the metal-insulator transition (MIT) for noninteracting electrons in the two-dimensional (2D) spin-orbit (symplectic) universality class. We find that the MF exponents near a boundary are different from those in the bulk. The exponents at a corner are found to be directly related to those at a straight boundary through a relation arising from conformal invariance. This provides direct numerical evidence for conformal invariance at the 2D spin-orbit MIT. The presence of boundaries modifies the MF of the whole sample even in the thermodynamic limit.

Figure 1

Systems studied: (a) A rhombus with the bulk, surface, and corner regions of sizes $l\times l$, $l\times h$, and $w\times w$ sites, correspondingly; (b) a cylinder with the bulk ($L\times l$) and surface ($L\times h$) regions.

Figure 2

PDFs of logarithm of WF amplitudes on tori (red), in the bulk region of cylinders (black), in the surface region of cylinders (blue), and in the corner region with $\theta =\pi /2$ (pink) and $\theta =\pi /4$ (light blue); $L=120$. Inset: Semilogarithmic plot.

Figure 3

(a) Bulk (red), surface (blue), corner $\theta =\pi /2$ (pink), and whole cylinder (orange) $f(\alpha )$ spectra, with error bars shown at integer values of $q$. Red, blue, and pink curves represent $f^x(4-\alpha )+\alpha -2$. Inset: Scaling plot of Eq. (8) at $q=1$ (filled circles) and $q=3$ (open circles). (b) Corner $f(\alpha )$ spectra at $\theta =\pi /4$ (light blue), $\pi /2$ (pink), and $3\pi /4$ (green), with error bars shown at integer $q$ (and at $q=-0.5$ for $\theta =\pi /4$). Curves represent the conformal relation (5). Inset: Numerical results for ${\alpha }_q^x$ compared with Eqs. (5) (colored curves).

Figure 4

(a) The numerical data for the bulk exponents ${\Delta }_q^b$ (red) are compared with their mirror image ${\Delta }_{1-q}^b$ (gray). The green line represents ${\alpha }_0^b-2$. (b) The exponents ${\Delta }_q/[q(1-q)]$ for the bulk (red), the surface (blue), and corners with $\theta =\pi /4$ (light blue), $\pi /2$ (pink), and $3\pi /4$ (green). The curves represent the theoretical prediction, Eq. (6). Inset: Bulk (red), surface (blue), and corner ($\theta =\pi /4$, light blue) exponents ${\tau }_q^x$, and ${\tau }_q^w$ for a whole rhombus with $\theta =\pi /4$ (black).