  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 rs the system is characterized by a single Fermi surface
with an out-of-plane spin polarization, while at slightly larger values of rs
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 work, 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 rs > 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.
