Using weak measurements to extract the $Z_2$ index of a topological insulator

TL;DR

This paper proposes a weak measurement protocol to determine the $Z_2$ index of a topological insulator, revealing universal responses in magnetization and charge measurements that are robust against disorder and interactions.

Contribution

It introduces a novel weak measurement scheme for extracting topological invariants in condensed matter systems, demonstrating universal responses in magnetization and charge measurements.

Findings

Weak value of magnetization responds universally to small magnetic fields.

Charge measurement setup yields a universal relation involving flux and capacitor charge.

Results are robust against disorder, interactions, and flux threading speed.

Abstract

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.