High-density limit of the two-dimensional electron liquid with Rashba spin-orbit coupling

We discuss by analytic means the theory of the high-density limit of the unpolarized two-dimensional electron liquid in the presence of Rashba or Dresselhaus spin-orbit coupling. A generalization of the ring-diagram expansion is performed. We find that in this regime the spin-orbit coupling leads to small changes of the exchange and correlation energy contributions, while modifying also, via repopulation of the momentum states, the noninteracting energy. As a result, the leading corrections to the chirality and total energy of the system stem from the Hartree-Fock contributions. The final results are found to be vanishing to lowest order in the spin-orbit coupling, in agreement with a general property valid to every order in the electron-electron interaction. We also show that recent quantum Monte Carlo data in the presence of Rashba spin-orbit coupling are well understood by neglecting corrections to the exchange-correlation energy, even at low density values.

Figure 1

Two different ways of occupying noninteracting chiral states in $\mathbf{k}$ space.

Figure 2

Numerical data (solid dots) from Ref. 25 for the total energy $\mathcal{E}(g,r_s,\chi )$ of Eq. (15), as functions of $\chi$ and at different values of $g$ and $r_s$. We obtain the solid lines by setting $\delta {\mathcal{E}}_{\mathrm{xc}}(g,r_s,\chi )=0$ in Eq. (15) and using the value of ${\mathcal{E}}_{\mathrm{xc}}(r_s)$ of Ref. 38. For reference, the empty dot in the top panel is the noninteracting energy without spin-orbit coupling (at $g=\chi =0$ and $r_s=1$).

Figure 3

Correction to the exchange energy (in Ry units at $r_s=1$) for the unpolarized state with Rashba spin-orbit coupling, at finite values of $\chi$. To obtain the value for generic $r_s$ one simply divides by $r_s$ [see Eq. (20)].

Figure 4

Plot of the second-order correlation energy for the noninteracting ground state, as function of $g$. The generalized chirality is $\chi ={\chi }_0(g)$ and the range of both plots is such that $g<\sqrt{2}$, which gives $\chi <1$. The top panel shows the direct term ${\mathcal{E}}_2^D(g,{\chi }_0(g))$ and the lower panel the exchange term ${\mathcal{E}}_2^X(g,{\chi }_0(g))$. The points represent numerical results from Monte Carlo integrations of Eqs. (26) and (28), and the solid lines serve as a guide for the eye.

Figure 5

Plot of the total second-order correlation energy ${\mathcal{E}}_2(g,{\chi }_0(g))$, with noninteracting occupation, as a function of $g$. The inset shows the region of small $g$, when $\chi <1$, and is obtained as the sum of the curves displayed in Fig. 4. The points represent numerical results, and the solid lines serve as a guide for the eye.