Examining density wave correlations in high pressure $\rm{La_3Ni_2O_7}$ through variational Monte Carlo

$\rm La_3Ni_2O_7$, a nickelate compound with a reported superconducting transition temperature of $\rm 80~K$, has attracted significant attention in recent years. Density-wave phenomena arising from strong electron correlations are widely regarded as key to unraveling the superconductivity mechanism, but the ordering and stability of these density waves remain a subject of contention in existing theoretical studies. In this work, we employ the variational Monte Carlo (VMC) method to thoroughly examine the nature of density waves as functions of Coulomb repulsion and exchange interactions in bilayer two-orbital model proposed for the high pressure phase of $\rm La_3Ni_2O_7$. We analyse the spin and charge correlation functions in a wide range of parameter space, and delineate a schematic phase diagram that separates different density-wave ground states. Our results provide useful insights into the understanding of electron correlations in $\rm La_3Ni_2O_7$, and highlight the potential of VMC to elucidate its superconducting mechanism.