Probing the $\nu=\frac{5}{2}$ quantum Hall state with electronic Mach-Zehnder interferometry

It was recently pointed out that Halperin's 113 topological order explains the transport experiments in the quantum Hall liquid at filling factor $\nu=5/2$. The 113 order, however, cannot be easily distinguished from other likely topological orders at $\nu=5/2$ such as the non-Abelian Pfaffian and anti-Pfaffian states and the Abelian Halperin 331 state in Fabry-Perot interferometry. We show that an electronic Mach-Zehnder interferometer provides a clear identification of these candidate $\nu=5/2$ states. Specifically, the $I$-$V$ curve for the tunneling current through the interferometer is more asymmetric in the 113 state than in other $\nu=5/2$ states. Moreover, the Fano factor for the shot noise in the interferometer can reach 13.6 in the 113 state, much greater than the maximum Fano factors of 3.2 in the Pfaffian and anti-Pfaffian states and 2.3 in the 331 state.