Phase diagram of the interacting persistent spin-helix state

We study the phase diagram of the interacting two-dimensional electron gas (2DEG) with equal Rashba and Dresselhaus spin-orbit coupling, which for weak coupling gives rise to the well-known persistent spin-helix phase. We construct the full Hartree-Fock phase diagram using a classical Monte Carlo method analogous to that used in Phys. Rev. B 96, 235425 (2017). For the 2DEG with only Rashba spin-orbit coupling, it was found that at intermediate values of the Wigner-Seitz radius $r_s$ the system is characterized by a single Fermi surface with an out-of-plane spin polarization, whereas at slightly larger values of $r_s$ it undergoes a transition to a state with a shifted Fermi surface and an in-plane spin polarization. The various phase transitions are first order, and this shows up in discontinuities in the conductivity, and the appearance of anisotropic resistance in the in-plane polarized phase. In this paper we show that the out-of-plane spin-polarized region shrinks as the strength of the Dresselhaus spin-orbit interaction increases and entirely vanishes when the Rashba and Dresselhaus spin-orbit coupling strengths are equal. At this point the system can be mapped onto a 2DEG without spin-orbit coupling, and this transformation reveals the existence of an in-plane spin-polarized phase with a single displaced Fermi surface beyond $r_s>2.01$. This is confirmed by classical Monte Carlo simulations. We discuss experimental observation and useful applications of the novel phase as well as caveats of using the classical Monte Carlo method.

Figure 1

Phase diagram of a 2D electron liquid with Rashba equal to Dresselhaus SOC in the HF approximation obtained by gauge transformation. It shows the results of an exact mapping to the case of no SOC followed by a HF calculation of the diagram. Here, $\tilde{\alpha }$ and $r_s$ are dimensionless measures for the strength of Rashba (Dresselhaus) SOC and electron-electron interactions, respectively. For the paramagnetic Fermi-liquid phases FL1 and FL2, there is no net spin polarization. The states of the PSH phase are spin-density waves analogous to the persistent spin helix. Only the state with in-plane polarization along $e_{\tilde{y}}$ has a uniform spin density.

Figure 2

Phase diagram of a2D electron liquid with Rashba SOC obtained by solving the HF equations using a MC method from Ref. [49]. Here, $\tilde{\alpha }$ and $r_s$ are dimensionless measures for the strength of Rashba SOC and electron-electron interactions, respectively: $\tilde{\alpha }$ corresponds to the ratio of the Fermi wavelength and spin-procession length, and $r_s$ is the Wigner-Seitz radius of the 2D electron system. The distinguishing ground-state features for each individual phase are indicated schematically. For the paramagnetic Fermi-liquid phase FL1 (FL2), there is no net spin polarization, and the ground state is a Fermi sea formed from one (both) spin subband(s). In contrast, the OP phase is characterized by a centered Fermi surface and an out-of-plane magnetization. The IP phase is the most unconventional, exhibiting an in-plane magnetization associated with a shifted Fermi sea.

Figure 3

The OP phase at $\tilde{\alpha }=0.3$ and $r_s=2.02$ in a 3D view. The OP phase comprises a single band with a circular Fermi surface and nontrivial out-of-plane spin polarization. Its in-plane spin polarization cancels out after summing over all the occupied states [49].

Figure 4

Phase diagram of a 2D electron liquid with equal Rashba and Dresselhaus SOCs, obtained by solving the HF equations using a MC simulation. It shows the result of a numerical HF calculation, which should give the same result as Fig. 1, but due to numerical uncertainty, instead shows also the possibility of the OP* phase. Here, $\tilde{\alpha }$ and $r_s$ are dimensionless measures for the strength of Rashba spin-orbit coupling and the electron-electron interactions, respectively: $\tilde{\alpha }$ corresponds to the ratio of the Fermi wavelength and spin-procession length, and $r_s$ is the Wigner-Seitz radius of the 2D electron system. The distinguishing features for each individual phase are indicated schematically by giving a picture that shows the spin texture. Each such picture is connected by an arrow to its point in the actual parameter space. For the Fermi-liquid phases FL1 and FL2, there is no net spin polarization. In contrast, the ${\mathrm{OP}}^{*}$ phase is characterized by an out-of-plane magnetization. However, it is believed to be an artifact of the finite size of the system. The IP phase exhibits an in-plane magnetization associated with a shifted Fermi sea and because of the SU(2) symmetry is equivalent to the PSH phase of Fig. 1.

Figure 5

The FL1 at $\tilde{\alpha }=\tilde{\beta }=0.81,r_s=1.65$ in a 3D view. The FL1 phase has two separated Fermi surfaces with in-plane spin polarization cancels out after sum over all occupied states.

Figure 6

The ${\mathrm{OP}}^{*}$ state at $\tilde{\alpha }=\tilde{\beta }=0.45,r_s=2.10$ in a 3D view. The ${\mathrm{OP}}^{*}$ phase has out-of-plane spin polarization, and the in-plane spin polarization cancels after sum over all occupied states.

Figure 7

The ${\mathrm{OP}}^{*}$ state at $\tilde{\alpha }=0.23,\tilde{\beta }=0.23,r_s=2.05$ in a 3D view. Compared with Fig. 6 with deceasing the SOC strength, the out-of-plane spin polarization becomes larger, and the Fermi surface is more compacted.

Figure 8

The IP state at $\tilde{\alpha }=0.23,\tilde{\beta }=0.23,r_s=2.30$ in a 3D view. The Fermi surface of the IP state with completely in-plane spin polarization is shifted along the $\frac{\pi }{4}$ direction.

Figure 9

Linear fit to find the ground state at $\tilde{\alpha }=\tilde{\beta }=0.08,r_s=2.10$, where $N$ is the lattice number we take in the MC simulation.