Precise ground state of multiorbital Mott systems via the variational discrete action theory

Determining the ground state of multiorbital Hubbard models is critical for understanding strongly correlated electron materials, yet, existing methods struggle to simultaneously reach zero temperature and infinite system size. The de facto standard is to approximate a finite dimension multiorbital Hubbard model with a $d=\ensuremath{\infty}$ version, which can then be formally solved via the dynamical mean-field theory (DMFT), although the DMFT solution is limited by the state of unbiased impurity solvers for zero temperature and multiple orbitals. The recently developed variational discrete action theory (VDAT) offers a new approach to solve the $d=\ensuremath{\infty}$ Hubbard model, with a variational ansatz that is controlled by an integer $\mathcal{N}$, and monotonically approaches the exact solution at an increasing computational cost. Here we propose a decoupled minimization algorithm to implement VDAT for the multiorbital Hubbard model in $d=\ensuremath{\infty}$ and study $\mathcal{N}=2\ensuremath{-}4$. At $\mathcal{N}=2$, VDAT rigorously recovers the multiorbital Gutzwiller approximation, reproducing known results. At $\mathcal{N}=3$, VDAT precisely captures the competition between the Hubbard $U$, Hund $J$, and crystal field $\mathrm{\ensuremath{\Delta}}$ in the two-orbital Hubbard model over all parameter space, with a negligible computational cost. For sufficiently large $U/t$ and $J/U$, we show that $\mathrm{\ensuremath{\Delta}}$ drives a first-order transition within the Mott insulating regime. In the large orbital polarization limit with finite $J/U$, we find that interactions have a nontrivial effect even for small $U/t$. VDAT will have far ranging implications for understanding multiorbital model Hamiltonians and strongly correlated electron materials.