Helical gaps in interacting Rashba wires at low electron densities

Rashba spin-orbit coupling and a magnetic field perpendicular to the Rashba axis have been predicted to open a partial gap (“helical gap”) in the energy spectrum of noninteracting or weakly interacting one-dimensional quantum wires. By comparing kinetic energy and Coulomb energy we show that this gap opening typically occurs at low electron densities where the Coulomb energy dominates. To address this strongly correlated limit, we investigate Rashba wires using Wigner crystal theory. We find that the helical gap exists even in the limit of strong interactions but its dependence on electron density differs significantly from the weakly interacting case. In particular, we find that the critical magnetic field for opening the gap becomes an oscillatory function of electron density. This changes strongly the expected signature of the helical gap in conductance measurements.

Figure 1

Single-particle spectra ${\epsilon }_{\pm }\left(k\right)$ for a weak magnetic field $\left(g{\mu }_{B}B<m{\alpha }_R^2\right)$. The color coding shows the spin orientation as a function of momentum. The corresponding density axis is shown on the right, and values of $B_{\mathrm{crit}}\left(\rho \right)$ for two densities are indicated.

Figure 2

Helical gap $\mathrm{\Delta }$ as a function of magnetic field $B$ and inverse electron density $\phi =1/\left(\rho {\ell }_{\mathrm{so}}\right)$. The surface plot and the green (bright) lines denote the numerical results obtained using DMRG analysis. The red line shows the critical magnetic field $B_{\mathrm{crit}}\left(\rho \right)$; see Eq. (10).

Figure 3

Critical magnetic field $B_{\mathrm{crit}}$ as a function of density $\rho$ for the noninteracting (red line) and the interacting (blue line) cases. In the noninteracting case, $B_{\mathrm{crit}}=0$ only at the critical density $\pi \rho =1/{\ell }_{\mathrm{so}}$. In the interacting case, in contrast, we find $B_{\mathrm{crit}}=0$ whenever the particle density is commensurate with the pitch of the effective spiral magnetic field. Examples for commensurate densities are shown in the right-hand panel, where the dots denote the electron positions and the spiral indicates the effective magnetic field; see Eq. (7).

Figure 4

Schematic plots of the conductance $G\left(\rho \right)$ for different values of $B$. For clarity, the lines for larger magnetic fields have been shifted downwards. In the interacting case (a), the conductance drops whenever the Wigner lattice is commensurate with the Rashba length. Moreover, the conductance saturates at $G_0$ towards low densities because $J\left(\rho \right)\to 0$, in stark contrast to the noninteracting case (b).

Figure 5

Different models for screening Coulomb interactions due to a metal at distance $d$ from an electron with charge $e$ in a nanowire along the $z$ direction: (a) infinite metallic surface, (b) metallic sphere, and (c) metallic cylinder.

Figure 6

Correction to the Coulomb potential due to screening by a metallic cylindrical gate with radius $R$ at distance $d$ to the wire.