Precise ground state of multi-orbital Mott systems via the variational discrete action theory

Determining the ground state of multi-orbital Hubbard models is critical for understanding strongly correlated electron materials, yet existing methods struggle to reach zero temperature and infinite system size simultaneously. Even in infinite dimensions, the solution via the dynamical mean-field theory (DMFT) is limited by the absence of unbiased impurity solvers for zero temperature and multiple orbitals. The recently developed variational discrete action theory (VDAT) offers a new approach, with a variational ansatz that is controlled by an integer \mathcal{N}, and monotonically approaches the exact solution at an exponentially increasing computational cost. Here we implement VDAT for the multi-orbital Hubbard model in d=\infty for \mathcal{N}=2-4. At \mathcal{N}=2, VDAT rigorously recovers the multi-orbital Gutzwiller approximation, reproducing known results. At \mathcal{N}=3, VDAT qualitatively and quantitatively captures the competition between U, J, and the crystal field in the two-band Hubbard model, with a negligible computational cost. VDAT will have far-ranging implications for understanding strongly correlated materials.