dc electrical current generated by upstream neutral modes

Quantum Hall phases are gapped in the bulk but support chiral edge modes, both charged and neutral. Here, we consider a circuit where the path from the source of electric current to the drain necessarily passes through a segment consisting solely of neutral modes. We find that upon biasing the source, a dc electric current is detected at the drain, provided there is backscattering between counterpropagating modes under the contacts placed in certain locations. Thus, neutral modes carry information that can be used to nonlocally reconstruct a dc charge current. Our protocol can be used to detect any neutral mode that counterpropagates with respect to all charge modes. Our protocol applies not only to the edge modes of a quantum Hall system, but also to systems that have neutral modes of nonquantum Hall origin. We conclude with a possible experimental realization of this phenomenon.

Figure 1

A single right-moving chiral charged mode (solid black line) represents the edge of a $\nu =1$ quantum Hall system extending above the figure, which is the “probe” system. Charges are injected at the source S and detected at the drain D. The “test” system extends below the figure, and has two counterpropagating neutral modes (dashed black lines), and possibly other charged chiral modes (dashed-dotted orange lines), which all have to be right moving for our scheme to be relevant. The edges of the two systems overlap only in regions II and IV, separated from the active region III by boundaries ${\mathrm{B}}_2,{\mathrm{B}}_3$. Density-density interactions between the chiral modes of the top and bottom systems exist only in regions II and IV, which also host the contacts ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$. Regions I and V are present to specify boundary conditions. Tunneling/scattering between the chiral modes occurs solely under the contacts. Reflection and transmission of a right-moving charge injected at S is shown schematically at ${\mathrm{B}}_3$, while a similar process for a left-moving neutral excitation is shown at ${\mathrm{B}}_2$.

Figure 2

The dc current at D as a function of the ${\lambda }_{12}$ and ${\lambda }_{23}$ for two different values of $v_1,v_2$, when ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are coupled.

Figure 3

A sheet of graphene in a perpendicular $B$ field is gated to have $\nu =1$ on the left and $\nu =0$ on the right. The central region of $\nu =0$ is overlain by an insulating ferromagnet, inducing the fully polarized phase of $\nu =0$ graphene in this region. The periphery of $\nu =0$ is presumed to be in the CAF state, with gapless Goldstone modes. The lowest subband of the radially confined Goldstone modes interacts with the $\nu =1$ edge mode and is detected by the scheme described in the text.