Energy density matrix formalism for interacting quantum systems: a quantum Monte Carlo study

TL;DR

This paper introduces an energy density matrix formalism for many-body quantum systems, enabling analysis of energy distribution and orbital energies, with applications demonstrated via quantum Monte Carlo simulations of electron gases and atoms.

Contribution

The paper develops a novel energy density matrix framework that parallels the 1RDM, providing new insights into energy distribution and quasiparticle-like spectra in correlated quantum systems.

Findings

Energy density matrix recovers exact spectrum for mean-field systems.

Occupation energies resemble quasiparticle energies in correlated systems.

Quantitative link between occupation energies and electron addition/removal energies.

Abstract

We develop an energy density matrix that parallels the one-body reduced density matrix (1RDM) for many-body quantum systems. Just as the density matrix gives access to the number density and occupation numbers, the energy density matrix yields the energy density and orbital occupation energies. The eigenvectors of the matrix provide a natural orbital partitioning of the energy density while the eigenvalues comprise a single particle energy spectrum obeying a total energy sum rule. For mean-field systems the energy density matrix recovers the exact spectrum. When correlation becomes important, the occupation energies resemble quasiparticle energies in some respects. We explore the occupation energy spectrum for the finite 3D homogeneous electron gas in the metallic regime and an isolated oxygen atom with ground state quantum Monte Carlo techniques implemented in the QMCPACK simulation code. The occupation energy spectrum for the homogeneous electron gas can be described by an effective mass below the Fermi level. Above the Fermi level evanescent behavior in the occupation energies is observed in similar fashion to the occupation numbers of the 1RDM. A direct comparison with total energy differences shows a quantitative connection between the occupation energies and electron addition and removal energies for the electron gas. For the oxygen atom, the association between the ground state occupation energies and particle addition and removal energies becomes only qualitative. The energy density matrix provides a new avenue for describing energetics with quantum Monte Carlo methods which have traditionally been limited to total energies.