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The recently developed variational discrete action theory (VDAT) provides asystematic variational approach to the ground state of the quantum many-bodyproblem, where the quality of the solution is controlled by an integer$\mathcal{N}$, and increasing $\mathcal{N}$ monotonically approaches the exactsolution. VDAT can be exactly evaluated in the $d=\infty$ multi-orbital Hubbardmodel using the self-consistent canonical discrete action theory (SCDA), whichrequires a self-consistency condition for the integer time Green's functions.Previous work demonstrates that $\mathcal{N}=3$ accurately capturesmulti-orbital Mott/Hund physics at a cost similar to the Gutzwillerapproximation. Here we employ a gauge constraint to automatically satisfy theself-consistency condition of the SCDA at $\mathcal{N}=3$, yielding an evenmore efficient algorithm with enhanced numerical stability. We derive closedform expressions of the gauge constrained algorithm for the multi-orbitalHubbard model with general density-density interactions, allowing VDAT at$\mathcal{N}=3$ to be straightforwardly applied to the seven orbital Hubbardmodel. We present results and a performance analysis using $\mathcal{N}=2$ and$\mathcal{N}=3$ for the $\textrm{SU}(2\textrm{N}_{\textrm{orb}})$ Hubbard modelin $d=\infty$ with $\textrm{N}_{\textrm{orb}}=2-8$, and compare to numericallyexact dynamical mean-field theory solutions where available. The developmentsin this work will greatly facilitate the application of VDAT at $\mathcal{N}=3$to strongly correlated electron materials.

A gauge constrained algorithm of VDAT at $\mathcal{N}=3$ for the multi-orbital Hubbard model