Missing Shapiro steps and the $8\pi$-periodic Josephson effect in interacting helical electron systems

Two-particle backscattering in time-reversal-invariant interacting helical electron systems can lead to the formation of quasiparticles with charge $e/2$. We propose a way to detect such states by means of the Josephson effect in the presence of proximity-induced superconductivity. In this case, the existence of $e/2$ charges leads to an $8\pi$-periodic component of the Josephson current which can be identified through measurement of Shapiro steps in Josephson junctions. In particular, we show that even when there is weak explicit time-reversal symmetry breaking, which causes the two-particle backscattering to be a subleading effect at low energies, its presence can still be detected in driven, current-biased Shapiro step measurements. The disappearance of some of these steps as a function of the drive frequency is directly related to the existence of non-Abelian zero-energy states. We suggest that this effect can be measured in current state-of-the-art Rashba wires.

Figure 1

(a) Shows the position of the chemical potential and the four low-energy linearized modes used to bosonize the system. (b) Shows the reconstruction of the band structure as a result of the RG-relevant interaction processes between these modes. Umklapp scattering opens a helical gap of size $g_{\mathrm{U}}$ between the two modes near $k=0$ and proximity-induced $s$-wave superconductivity opens gaps for the outer modes at $k=\pm 2m{\alpha }_R$ of size $g_{\mathrm{SC}}$, resulting in a fully gapped system.

Figure 2

A plot of single-particle spectrum of a Rashba spin-orbit coupled nanowire showing the two lowest transverse subbands, whose spin character as a function of momentum is shown by the color coding. Virtual transitions between subbands come with a spin flip, and electron-electron interactions allow us to couple states in different subbands. The inset shows the exact process responsible for spin-umklapp scattering, where the virtual transition between subbands (denoted by $g$) changes the spin character of two incoming particles, and then the four-particle interaction vertex (denoted by $V$) couples these, resulting in a process in the lowest subband where two spin-$\uparrow$ particles are scattered into two spin-$\downarrow$ particles.

Figure 3

Experimental setup for the measurement of the $8\pi$-periodic Josephson effect. Two superconductors underneath a Rashba wire are held at a phase difference $\varphi$. The blue portions of the wire are gapped by superconductivity, whereas the gray section is gapped by umklapp scattering. Fractionally charged bound states are indicated by red spheres.

Figure 4

The equivalent circuit diagram for our current-biased Josephson junction arrangement. The combination of the constant current source with a small ac offset provided by microwave irradiation of the junction gives the total current source. The transport through the junction divides into a tunnel current $I\left[\phi \right(t\left)\right]$ from tunneling of quasiparticles through the junction, and a resistive current in parallel which comes from transport through unproximitized above-gap states of the nanowire.

Figure 5

Numerical solutions to Eq. (19) with a dc bias current (${\alpha }_1=0$), showing different periodicity behavior depending on the amplitude of the dc current. We have made the representative choice ${\alpha }_m=\frac{1}{10}$ and ${\alpha }_p=\frac{1}{100}$ which is consistent with our expectations of the relative size of the contributions to the Josephson current from Majorana fermions and parafermions (see main text). (a) Shows the superconducting phase changing in $2\pi$ steps for the choice ${\alpha }_0=1.5$, whereas (b) is plotted for ${\alpha }_0=1.1$, and shows residual small steps at $2\pi$ and dominant steps at $4\pi$ which correspond to the dominant transport being via tunneling of Majorana fermions. In (c), we choose ${\alpha }_0=1.08$, and recover jumps of $8\pi$ in the superconducting phase due to tunneling of parafermions.

Figure 6

A plot of the current-voltage relationship for the system driven with a small amplitude ac current with a large dc offset (with corresponding magnitudes ${\alpha }_0=1.08+\delta I$ and ${\alpha }_1=0.01$). The excess current $\delta I$ over the critical value at zero drive is plotted on the vertical axis. The Josephson junction has a dominant $2\pi$-periodic component, and subleading $4\pi$- and $8\pi$-periodic components (${\alpha }_m=\frac{1}{10}$ and ${\alpha }_p=\frac{1}{100}$). (a) At the fastest drive frequency (comparable to that of the $2\pi$-periodic component of the Josephson current) shown in the top curve, we find all integer Shapiro steps. As we drive at a slower frequency (middle curve), the first Shapiro step shrinks, and then disappears, leaving only Shapiro steps which are multiples of 2. (b) Driving at smaller frequencies, we find a double-step structure (top curve), but the size of the intermediate steps shrink as we reduce $\tilde{\omega }$ (middle curve). Eventually, at the lowest frequencies, we see a $4n\omega$ step structure. Note that the extra steps at $V\approx 2\omega /3$ in the top two lines in (a) result from the fact that we are driving nonadiabatically, but are not relevant to experimental measurements as they do not occur at integer multiples of the driving frequency. Note that the I-V curves have been offset in the y-direction to make the step structure easily visible.