Using weak measurements to extract the Z2 index of a topological insulator

Recently, there has been an interest in applying the concept of weak values and weak measurements to condensed matter systems. Here a weak measurement protocol is proposed for obtaining the Z2 index of a topological insulator. The setup consists of a topological insulator with a hole pierced by a time dependent Aharonov-Bohm flux. A certain weak value (Ags) associated with the time-integrated magnetization in the hole has a universal response to a small ambient magnetic field (B), namely, AgsB=2. This result is unaffected by disorder, interactions, and to a large extent, the speed of the flux threading. It hinges mainly on preventing the flux from leaking outside the hole, as well as being able to detect magnetization at a resolution of a few spins. A similar result may be obtained using only charge measurements, in a setup consisting of a double quantum dot weakly coupled to an LC circuit. Here one obtains (φ)weakQ0=2, where (φ)weak is a weak value associated with the flux on the inductor and Q0 is the average capacitor charging. The universality of these results suggests that they may be used as a test bed for weak values in condensed matter physics.