Renormalization of anticrossings in interacting quantum wires with Rashba and Dresselhaus spin-orbit couplings

We discuss how electron-electron interactions renormalize the spin-orbit induced anticrossings between different subbands in ballistic quantum wires. Depending on the ratio of spin-orbit coupling and subband spacing, electron-electron interactions can either increase or decrease anticrossing gaps. When the anticrossings are closing due to a special combination of Rashba and Dresselhaus spin-orbit couplings, their gap approaches zero as an interaction dependent power law of the spin-orbit couplings, which is a consequence of Luttinger liquid physics. Monitoring the closing of the anticrossings allows us to directly measure the related renormalization group scaling dimension in an experiment. If a magnetic field is applied parallel to the spin-orbit coupling direction, the anticrossings experience different renormalizations. Since this difference is entirely rooted in electron-electron interactions, unequally large anticrossings also serve as a direct signature of Luttinger liquid physics. Electron-electron interactions furthermore increase the sensitivity of conductance measurements to the presence of anticrossing.

Figure 1

The proposed experimental setup consists of a two-dimensional electron gas (2DEG), in which the electrons are confined to a quasi-one-dimensional wire region (red). The confinement is due to electrostatic gates placed atop the 2DEG (green), and depleting it outside the wire region (light yellow). By modulating the electrostatic potential of the gates, the width of the wire can be changed in situ. The electron density inside the wire can be controlled by an additional gate (not shown).

Figure 2

Band structure of the two-subband quantum wire in the noninteracting case and without interband couplings (the labels $i,\sigma$ indicate subband index and spin polarization). The spectra are calculated using Eqs. (1) and (2) after setting $U=0$, $\alpha +\beta =0$, and $\alpha -\beta ={\alpha }_0$. The red circles highlight the intersubband crossings that are lifted in the presence of intersubband spin-orbit coupling. (a) Band structure calculated for a large energy difference of ${\delta }_{12}=0.8$ meV between the bottoms of the first and second subbands, corresponding to a situation in which the intersubband spin-orbit coupling induces forward scattering (FS). (b) The case where the crossings occur at the bottoms of the second bands (with an offset of ${\delta }_{12}\approx 0.4$ meV). Finally, (c) indicates that small band offsets, here ${\delta }_{12}=0.2$ meV, lead to interband backscattering (BS). Higher unoccupied bands are neglected.

Figure 3

Band structure of the two-subband quantum wire in the noninteracting case for relatively large interband couplings (all parameters are chosen as in Fig. 2, except for $\alpha +\beta =0.5{\alpha }_0$). The band offsets are again given by (a) ${\delta }_{12}=0.8$ meV, (b) ${\delta }_{12}\approx 0.4$ meV, and (c) ${\delta }_{12}=0.2$ meV.

Figure 4

Ratio of the Fermi velocities $v_{F2}/v_{F1}$ at the crossing points (see Fig. 2). The dashed vertical line indicates the interband offset at which the band crossings occur at the bottom of the second band. For smaller ${\delta }_{12}$, the intersubband spin-orbit coupling gives rise to backscattering (BS), while larger values of ${\delta }_{12}$ lead to forward scattering (FS). The dotted lines delimit the excluded regime in which strong interactions in the second subband eventually modify the physics (we choose $K_{2c}=0.5$ as the criterion to distinguish weak from strong interactions).

Figure 5

Luttinger liquid parameters $K_{ic}$ in the charge sector of band $i=1,2$ if the chemical potential intersects the crossing points of the spectra shown in Fig. 2, as a function of the band offset ${\delta }_{12}$. For different band offsets, the crossing point occurs at different points of the band structure, which alters the Fermi velocities. This in turn modifies the value of the Luttinger liquid parameters. Like in Fig. 4, the dotted vertical lines (and the dotted horizontal line) indicate $K_{2c}=0.5$, the dashed vertical line indicates the transition point between forward (FS) and backscattering (BS).

Figure 6

The parameter ($1-g_{12}$) as a function of the band offset ${\delta }_{12}$ ($g_{12}$ is the RG scaling dimension of the intersubband spin-orbit couplings). As in Fig. 4, the vertical dashed line marks the transition between forward (FS) and backscattering (BS), the vertical dotted lines indicate the regime in which the band crossing occurs close to the bottom of the upper bands.

Figure 7

Band structure of the two subband quantum wire for a band offset of ${\delta }_{12}=0.1$ meV, $\alpha -\beta ={\alpha }_0$, and $\alpha +\beta =0.1{\alpha }_0$. (a) The unrenormalized band structure, and (b) the one renormalized by electron-electron interactions.

Figure 8

Scaling of the intersubband anticrossings as a function of the bare intersubband interaction $\alpha +\beta$ for fixed $\beta =-{\alpha }_0/2$. The unrenormalized gap depends linearly on $\alpha +\beta$, while the renomormalized one obeys the power law defined in Eq. (13). As an example, this graph contrasts the renormalized and unrenormalized anticrossing gaps for ${\delta }_{12}=0.2$, in which case the anticrossing gap scales with the exponent $1/(2-g_{12})\approx 0.91$ as function of the bare intersubband spin-orbit coupling. The dotted line marks an energy of $0.005\text{meV}\equiv 50\text{mK}$, which constitutes a realistic experimental measurement resolution.

Figure 9

Band structure of the two subband quantum wire in the noninteracting case, without interband couplings, and with an applied magnetic field parallel to the spin quantization axis set by the intrasubband spin-orbit coupling (we use $U=0$, $\alpha +\beta =0$, and $\alpha -\beta ={\alpha }_0$ for ${\delta }_{12}=0.2\text{meV}$ and a Zeeman energy of $\pm 0.2\text{meV}$ for spin up and spin down). For this value of the magnetic field, crossing between the bands $1,\downarrow$ and $2,\uparrow$ corresponds to forward scattering, while the one between $1,\uparrow$ and $2,\downarrow$ corresponds to backscattering.

Figure 10

Band structure of the two subband quantum wire in the noninteracting case, without interband couplings, and with an applied magnetic field perpendicular to the spin quantization axis set by the intrasubband spin-orbit coupling (we use $U=0$, $\alpha +\beta =0$, and $\alpha -\beta ={\alpha }_0$ for ${\delta }_{12}=0.2\text{meV}$ and ${\Delta }_B=0.05\text{meV}$). The dotted lines indicate the bands without magnetic field.