Metal-insulator and magnetic phase diagram of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ from auxiliary field quantum Monte Carlo and dynamical mean field theory

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 ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$. Both methods predict a low-temperature, pressure-driven metal-insulator transition accompanied by a ferromagnetic-antiferromagnetic transition. The properties of the ferromagnetic state vary nonmonotonically 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 in the details of 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.

Figure 1

Crystal structure of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ (central portion), with calcium shown in blue, oxygen in gray, and ruthenium in red. The primitive vectors $\vec{a}, \vec{b}$, and $\vec{c}$ are defined using the orthorhombic convention. Left side: Depiction of the short-bond ${\mathrm{RuO}}_6$ octahedron occurring in the low-$T$, ambient-pressure S-Pbca structure (top) with a depiction of the dominant Ru multiplet term in the local electronic structure of the insulating state beneath. The $d_{xy}$ orbital (red bar) is fully occupied and the $xz/yz$ orbitals (blue bars) are half-filled and in a high-spin state. Right side: Depiction of the long-bond ${\mathrm{RuO}}_6$ octahedron occurring in the high-$T$ or high-$P$ L-Pbca structures (top), with a depiction of the dominant Ru multiplet term in the ferromagnetic state beneath. A nearly half-filled, spin-polarized $xy$ orbital (red bar) with the remaining hole is in a superposition of $xz/yz$ states (blue bars).

Figure 2

Comparison of the charge gap predictions from DMFT and AFQMC for the L-Pbca and S-Pbca structures. DMFT results are calculated at 60 K and AFQMC results are calculated at 0 K. Left: Spectral functions per Ru obtained using DMFT simulations with analytical continuation for the S-$295\mathrm{K}$ structure (a) and L-$5\mathrm{GPa}$ structure (c). Right: Extrapolation of the charge gap to the thermodynamic limit in AFQMC calculations of the S-$295\mathrm{K}$ structure (b) and L-$5\mathrm{GPa}$ structure (d). In (b) and (d), the $M$ on the $x$ axis represents the number of unit cells used in the AFQMC simulations.

Figure 3

Comparison of magnetic moment distributions and orbital occupancies obtained by DMFT at 60 K and AFQMC at 0 K for the experimentally acquired L-Pbca L-$5\mathrm{GPa}$ structure. (a) Magnetic moment distributions from DMFT: $60\%$ of the magnetic moment is in the $d_{xy}$ band. (b) Magnetic moment distributions from AFQMC: $68\%$ of the magnetic moment is in the $d_{xy}$ band. (c) Orbital occupancies from DMFT and AFQMC: the $y$ axis is shifted by an average density of $4/3$. A smaller $d_{xy}$ band occupancy results from a negative crystal field splitting.

Figure 4

(a) Phase diagram of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ for different lattice structures. Both DMFT and AFQMC predict the S-$295\mathrm{K}$ structure to be an AFM-I and the L-$5\mathrm{GPa}$ structure to be an FM-M. At intermediate pressures, both DMFT and AFQMC find ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ to be a metal, but AFQMC predicts an FM-M state, while DMFT predicts a paramagnetic metal (PM-M) state (based on the range of temperatures we could access). (b) The red line depicts the ground-state magnetic moments obtained using AFQMC as a function of the pressure. The green line is a guide for the eye that denotes the critical temperature obtained as a function of the pressure with DMFT. Pressures at which there is no FM order are depicted as having negative critical temperatures. Note that the structure at $P=6\mathrm{GPa}$ shown in (b) is obtained by linear extrapolation of experimentally determined atom positions at $P=1\mathrm{GPa}$ and $P=3\mathrm{GPa}$. (See Ref. [38], Sec. III B, for details regarding the interpolations).