Superuniversality of topological quantum phase transition and global phase diagram of dirty topological systems in three dimensions

Pallab Goswami, Sudip Chakravarty

Published 2016-03-11Version 1

The quantum phase transition between two clean, non interacting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For dirty systems, the notion of a sharp spectral gap is invalidated by the disorder induced sub-gap states and their localization is essential for describing topological and trivial insulators. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point controls a direct quantum phase transition between two topologically distinct localized phases, with a dynamic scaling exponent $z=1$ and a localization length exponent $\nu_M=1$, implying the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase, allowing for only localization-delocalization (metal-insulator) transitions. In the global phase diagram the disorder controlled fixed point where superuniversality disappears actually serves as a multicritical point, where two localized and the delocalized diffusive phases meet. The nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. For particle hole symmetric disorders, we argue that the localization length exponent at the multicritical point is $\nu_M =2/3$, which saturates the bound provided by Chayes-Chayes-Fisher-Spencer theorem.