Conformal Invariance in Two-Dimensional Electron Systems with Quenched Disorder
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The universal critical properties at conventional continuous phase transitions in two dimensions are well understood by conformal field theories (CFTs). In contrast, for continuous phase transitions driven by quenched disorder, field theories that have a power of predicting critical properties are still missing. A localization-delocalization transition of a non-interacting electron in disorder potential, i.e., Anderson transition, is one such example of a disorder-driven quantum phase transition lacking satisfying theoretical description. In this work we present direct evidence for the presence of conformal invariance at the Anderson transition in two dimensions by verifying relations derived from the boundary CFT 1. This can be viewed as a first step towards establishing a conformal field theoretical description of the Anderson transition in two dimensions.
At critical points of the Anderson transition, wave functions possess multifractality, and the inverse participation ratio, $dr |\psi(r)|^2q$, shows a power-law dependence on the system size with a non-trivial exponent $\Delta_q$ called anomalous dimensions. If we suppose that $|\psi(\boldmathr)|^2q$ is represented by a local operator in an underlying CFT and that the scaling dimension of this operator is $\Delta_q$, then the boundary CFT analysis of these local operators on the $z$-plain and the $w$-plain in Fig.1 leads to relations between surface anomalous exponents multifractality along a straight boundary as shown in Fig.1(a) and corner anomalous exponents multifractaliy at a corner with the wedge angle $þeta$ as shown in Fig.1(b).
We have checked the validity of these relations numerically for a critical point of a two-dimensional tight-binding model of non-interacting electrons with quenched disorder and spin-orbit interactions. From one-point correlation functions of $|\psi(r)|^2q$, anomalous dimensions $\Delta_q$ and multifractal singular spectra $f(\alpha)$ of surface and corner regions are calculated. The surface and corner multifractal exponents were found to be related as expected. Furthermore, we calculated the exponents of power-law decaying two-point correlation functions $|\psi(\boldmathr_1)|^2q|\psi(r_2)|^2q$ for the two geometries shown in Fig. 1(a) and (b). We found that these exponents are also related in the way predicted by the boundary CFT. These results provide direct evidence for conformal invariance and for the primary nature of these operators at a critical point of the Anderson transition.
1) H. Obuse, A.R. Subramaniam, A. Furusaki, I.A. Gruzberg, and A.W.W. Ludwig, to appear in Phys. Rev. Lett. ; cond-mat/0609161.