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

TL;DR

The paper introduces off-shell effective energy theory (OET), a formalism for thermodynamic descriptions of quantum Hamiltonians, demonstrated on the Hubbard model across different dimensions, showing accurate results with low computational cost.

Contribution

The paper presents OET and CPE as novel methods for analyzing lattice models, providing a unified approach applicable from one to infinite dimensions.

Findings

OET accurately describes the Hubbard model in various dimensions.

CPE provides a practical approximation with negligible computational cost.

OET captures different physical regimes like Fermi liquids and Mott insulators.

Abstract

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.