Invertibility of the retarded response functions for initial mixed states: application to one-body reduced density matrix functional theory

TL;DR

This paper generalizes the conditions for the kernel of retarded response functions from pure to mixed states, especially for thermodynamic ensembles, revealing that potentials in the kernel must commute with the initial Hamiltonian under certain assumptions.

Contribution

It extends the characterization of the kernel of retarded response functions to mixed states, providing conditions applicable to thermodynamic ensembles and simplifying the kernel analysis.

Findings

Potentials in the kernel commute with the initial Hamiltonian under certain conditions.

Conditions are more restrictive for thermodynamic ensembles than for pure states.

The kernel of the response function is less complex for the one-body reduced density matrix in thermodynamic ensembles.

Abstract

In [J. Chem. Phys. 143, 054102 (2015)] I have derived conditions to characterize the kernel of the retarded response function, under the assumption that the initial state is a ground state. In this article I demonstrate its generalization to mixed states (ensembles). To make the proof work, the weights in the ensemble need to be decreasing for increasing energies of the pure states from which the mixed state is constructed. The resulting conditions are not easy to verify, but under the additional assumptions that the ensemble weights are directly related to the energies and that the full spectrum of the Hamiltonian participates in the ensemble, it is shown that potentials only belong to the kernel of the retarded response function if they commute with the initial Hamiltonian. These additional assumptions are valid for thermodynamic ensembles, which makes this result also physically relevant. The conditions on the potentials for the thermodynamic ensembles are much stronger than in the pure state (zero temperature) case, leading to a much less involved kernel when the conditions are applied to the retarded one-body reduced density matrix response function.