  Layered perovskite ruthenium oxides exhibit a striking series of
metal-insulator and magnetic-nonmagnetic phase transitions easily tuned by
temperature, pressure, epitaxy, and nonlinear drive. In this work, we combine
results from two complementary state of the art many-body methods, Auxiliary
Field Quantum Monte Carlo and Dynamical Mean Field Theory, to determine the
low-temperature phase diagram of Ca$_2$RuO$_4$. Both methods predict a low
temperature, pressure-driven metal-insulator transition accompanied by a
ferromagnetic-antiferromagnetic transition. The properties of the ferromagnetic
state vary non-monotonically with pressure and are dominated by the ruthenium
$d_{xy}$ orbital, while the properties of the antiferromagnetic state are
dominated by the $d_{xz}$ and $d_{yz}$ orbitals. Differences of detail in the
predictions of the two methods are analyzed. This work is theoretically
important as it presents the first application of the Auxiliary Field Quantum
Monte Carlo method to an orbitally-degenerate system with both Mott and Hunds
physics, and provides an important comparison of the Dynamical Mean Field and
Auxiliary Field Quantum Monte Carlo methods.
