Off-shell effective energy theory: a unified treatment of the Hubbard model from d=1 to d= ∞

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. 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 present dual realizations of OET based on a partitioning of the kinetic and potential energy, denoted as K and X, respectively. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞ , showing favorable agreement with exact or state-of-the-art results over all parameter space; and a negligible computational cost. Physically, K describes the Fermi liquid, while 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.