  Here we propose an exact formalism, off-shell effective energy theory (OET),
which provides a thermodynamic description of a generic quantum Hamiltonian.
The OET is based on a partitioning of the Hamiltonian and a corresponding
density matrix ansatz constructed from an off-shell extension of the
equilibrium density matrix; and there are dual realizations based on a given
partitioning. To approximate OET, we introduce the central point expansion
(CPE), which is an expansion of the density matrix ansatz, and we renormalize
the CPE using a standard expansion of the ground state energy. We showcase the
OET for the one band Hubbard model in d=1, 2, and $\infty$, using a
partitioning between kinetic and potential energy, yielding two realizations
denoted as $\mathcal{K}$ and $\mathcal{X}$. OET shows favorable agreement with
exact or state-of-the-art results over all parameter space, and has a
negligible computational cost. Physically, $\mathcal{K}$ describes the Fermi
liquid, while $\mathcal{X}$ gives an analogous description of both the
Luttinger liquid and the Mott insulator. Our approach should find broad
applicability in lattice model Hamiltonians, in addition to real materials
systems.
