  Here we propose the Variational Discrete Action Theory (VDAT) to study the
ground state properties of quantum many-body Hamiltonians. VDAT is a
variational theory based on the sequential product density matrix (SPD) ansatz,
characterized by an integer $\mathcal{N}$, which monotonically approaches the
exact solution with increasing $\mathcal{N}$. To evaluate the SPD, we introduce
a discrete action and a corresponding integer time Green's function. We use
VDAT to exactly evaluate the SPD in two canonical models of interacting
electrons: the Anderson impurity model (AIM) and the $d=\infty$ Hubbard model.
For the latter, we evaluate $\mathcal{N}=2-4$, where $\mathcal{N}=2$ recovers
the Gutzwiller approximation (GA), and we show that $\mathcal{N}=3$, which
exactly evaluates the Gutzwiller-Baeriswyl wave function, provides a truly
minimal yet precise description of Mott physics with a cost similar to the GA.
VDAT is a flexible theory for studying quantum Hamiltonians, competing both
with state-of-the-art methods and simple, efficient approaches all within a
single framework.
