Disordered Weak and Strong Topological Insulators

Abstract

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or "defeated" in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents.PACS numbers: 73.20.Fz, 71.30.+h Robustness against disorder is a defining property of the topological quantum phenomena. Depending on the degree of this robustness, three-dimensional (3D) Z 2 topological insulators (TIs) [1][2][3] are classified into strong and weak (STI and WTI). Bulk-surface correspondence implies that an STI exhibits a single helical Dirac cone that is protected, while a WTI manifests generally an even number (possibly null) of Dirac cones depending on the orientation of the surface [4].Unusual robustness of Dirac electrons (especially in the case of a single Dirac cone) against disorder has been widely recognized in the study of graphene [5,6]. As a consequence of the absence of backward scattering [7], the Dirac electrons do not localize. However, in the presence of valleys (even number of Dirac cones) they do localize mediated by intervalley scatterings [8]. Does this mean that an STI continues to be an STI in the presence of arbitrarily strong disorder, while a WTI simply collapses on the switching on of the short-ranged potential disorder that induces intervalley scattering?Recent studies on the disordered WTI [9,10] seem to suggest that the reality is much different. Our global phase diagram depicted in Fig. 1 finds its way also in this direction. This phase diagram is established by a combination of the study of the averaged two-terminal conductance and of the quasi-1D decay length in the transfer matrix approach. In the actual computation the 3D disordered Z 2 topological insulator is modeled as an WilsonDirac-type tight-binding Hamiltonian with an effective (k-dependent) mass term m(k) = m 0 + m 2 µ=x,y,z (1 − cos k µ ) [11], implemented on a cubic lattice. The topological nature of the model is controlled by the ratio of two mass parameters m 0 and m 2 such that an STI phase with Z 2 (one strong and three weak) indices [4] (ν 0 , ν 1 ν 2 ν 3 ) = (1, 000) appears when −2 < m 0 /m 2 < 0, while the regime of parameters −4 < m 0 /m 2 < −2 falls on a WTI phase with (ν 0 , ν 1 ν 2 ν 3 ) = (0, 111) [12].The obtained "global" phase diagram depicted in Fig. 1 highlights the main results of the Letter. This phase diagram shows how disorder modifies the above topological classification in the clean limit (naturally as a function of the strength of disorde...

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