${\mathbb{Z}}_2$ Peak of Noise Correlations in a Quantum Spin Hall Insulator

We investigate the current noise correlations at a quantum point contact in a quantum spin Hall structure, focusing on the effect of a weak magnetic field in the presence of disorder. For the case of two equally biased terminals we discover a robust peak: the noise correlations vanish at $B=0$ and are negative for $B\ne 0$. We find that the character of this peak is intimately related to the interplay between time reversal symmetry and the helical nature of the edge states and call it the ${\mathbb{Z}}_2$ peak.

Figure 1

Six terminal Hall bar structure with a quantum point contact. VP 1 and VP 2 are voltage probes which allow for inelastic scattering.

Figure 2

Scattering model for the structure shown in Fig. 1 in the presence of a magnetic field and disorder. The setup is partitioned so that the overall scattering matrix can be obtained from combining the transfer matrices (denoted by $M$). A magnetic field causes a flux $\Phi$ in the QPC region and nonzero backscattering along each arm of the helical edge states. Disorder, modeled by random scattering phases ${\xi }_i$, ${\phi }_u$ and ${\phi }_l$ as indicated in the figure, leads to a random distribution of the overall scattering matrices.

Figure 3

Probability distributions of the cross-correlation noise power $P_{34}$ as a function of magnetic field $B$, obtained from the scattering model with 100 000 disordered samples in the upper panel, and from numerical simulations with 1200 disordered samples in the lower panel. Overlaying both panels, the solid lines show the mean of $P_{34}$, clearly showing the ${\mathbb{Z}}_2$ peak at $B=0$. The dashed lines indicate the average of $P_{34}$ evaluated for a circular unitary ensemble of scattering matrices.