  In the study of strongly-correlated, many-electron systems, the Hubbard
Kanamori (HK) model has emerged as one of the prototypes for transition metal
oxide physics. The model is multi-band in nature and contains Hund's coupling
terms, which have pronounced effects on metal-insulator transitions,
high-temperature superconductivity, and other physical properties. In the
following, we present a complete theoretical framework for treating the HK
model using the ground state Auxiliary Field Quantum Monte Carlo (AFQMC) method
and analyze its performance on few-band models whose parameters approximate
those observed in ruthenate, rhodates, and other materials exhibiting Hund's
physics. Unlike previous studies, the constrained path and phaseless
approximations are used to respectively control the sign and phase problems,
which enables high accuracy modeling of the HK model's ground state properties
within parameter regimes of experimental interest. We demonstrate that, after
careful consideration of the Hubbard-Stratonovich transformations and trial
wave functions employed, relative errors in the energy of less than 1% can
routinely be achieved for moderate to large values of the Hund's coupling
constant. Crucially, our methodology also accurately predicts magnetic ordering
and phase transitions. The results presented open the door to more predictive
modeling of Hund's physics within a wide range of strongly-correlated materials
using AFQMC.
