Spin-orbit coupling in quasi-one-dimensional Wigner crystals

We study the effect of Rashba spin-orbit coupling (SOC) on the charge and spin degrees of freedom of a quasi-one-dimensional (quasi-1D) Wigner crystal. As electrons in a quasi-1D Wigner crystal can move in the transverse direction, SOC cannot be gauged away in contrast to the pure 1D case. We show that for weak SOC, a partial gap in the spectrum opens at certain ratios between the density of electrons and the inverse Rashba length. We present how the low-energy branch of charge degrees of freedom deviates due to SOC from its usual linear dependence at small wave vectors. In the case of strong SOC, we show that the spin sector of a Wigner crystal cannot be described by an isotropic antiferromagnetic Heisenberg Hamiltonian anymore and that instead the ground state of neighboring electrons is mostly a triplet state. We present a new spin sector Hamiltonian and discuss the spectrum of a Wigner crystal in this limit.

Figure 1

The arrangement of electrons which are strongly confined in the $Z$ direction, but more weakly confined in the $Y$ direction by a harmonic potential, leading to the zigzag form of the Wigner crystal. The unit cell of the zigzag state is shown in orange and contains two distinct lattice sites labeled 1 and 2. We model the screening of the long-range part of the Coulomb interaction between electrons seen in real experimental systems by means of a metallic gate at a distance $d$ below the confined electrons (blue circles). The presence of this gate results in image charges (green circles) which cause the long-range part of the Coulomb interaction to decay as $1/{\left|X\right|}^3$ as expected for a dipole potential.

Figure 2

The eigenvalues of the matrix $V_k$ as a function of $k$ for the four charge-sector normal modes in the zigzag Wigner crystal without SOC present. Whereas the exact parameters for this plot are given in the text, the qualitative behavior of these eigenvalues does not depend sensitively on their values. Focusing on the lowest mode, we see that this eigenvalue, which corresponds to the frequency squared of the lowest in energy oscillator is approximately quadratic for small $k$, leading to a linear dispersion relation for this mode.

Figure 3

The spectrum of the lowest branch of charge degrees of freedom with SOC. Here $\sqrt{{({\epsilon }_1-{\epsilon }_2)}^2-{\lambda }^2}$ is in units of $e^2/\left(\epsilon a\right)$, and $k$ is in units of $1/a$. The parameters are presented in the text. Here again, as for Fig. 2, the qualitative behavior of $\sqrt{{({\epsilon }_1-{\epsilon }_2)}^2-{\lambda }^2}$ is important. We see that $\sqrt{{({\epsilon }_1-{\epsilon }_2)}^2-{\lambda }^2}$ noticeably deviates from linear dependence for small $k$.