Using weak measurements to extract the $Z_2$ 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 $Z_2$ 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 ($A_{gs}$) associated with the time-integrated magnetization in the hole has a universal response to a small ambient magnetic field ($B$), namely $A_{gs}B = 2 \hbar$. 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 $\langle \phi \rangle_{weak} Q_0 =2\hbar$, where $\langle \phi \rangle_{weak}$ is a weak value associated with the flux on the inductor and $Q_0$ is the average capacitor charging. The universality of these results suggests that they may be used as a testbed for weak values in condensed matter physics.