Multifractality at the metal-quantum spin Hall insulator transition in two dimensions
The quantum spin-Hall (QSH) effect occurs in a new kind of a topological insulator characterized by the $Z_2$ topological number. Since QSH system possesses time-reversal symmetry and broken spin-rotation symmetry, this system is naively expected to belong to the symplectic class if taking account only of symmetries but ignoring the topological nature.
We investigate various critical properties at the Anderson transition of the QSH system in two-dimensions (2D) based on the network model. In Ref. , we show that the critical exponent characterizing the divergence of the localization length at criticality is identified with that of the ordinary symplectic class in 2D.
We also investigate bulk and boundary multifractality in the QSH system . When reflecting boundaries are imposed on the QSH network model, there exist two kinds of critical points depending on whether a boundary induces edge states in the adjacent insulating phase. We found that bulk multifractality in the QSH system are same as that of the ordinary symplectic class in 2D. It is also clarified that boundary multifractality at the critical point of the metal - ordinary insulator (absence of edge states) transition is same as that of the ordinary symplectic class, while boundary multifractality at the critical point of the metal-QSH insulator (presence of edge states) transition is completely different from that of the ordinary symplectic class. Therefore, boundary multifractality observed at the latter critical point is considered as a new boundary critical phenomenon in the symplectic class, reflecting the presence of topologically non-trivial edge states in the adjacent insulating phase. This is the first example for the presence of different boundary multifractality in the same universality class.
 HO, A. Furusaki, S. Ryu, and C. Mudry, PRB 76, 075301 (2007)  HO, A. Furusaki, S. Ryu, and C. Mudry, PRB 78, 115301 (2008)