Continuum Mechanics for Quantum Many-Body Systems: The Linear Response Regime

TL;DR

This paper introduces a continuum mechanics approach to quantum many-body systems, deriving a linear equation of motion for the current density that simplifies analysis in the linear response regime and remains computationally manageable.

Contribution

It presents a novel continuum mechanics framework for quantum systems, replacing orbital-based methods with a collective field approach and deriving a hermitian eigenvalue problem for excitation energies.

Findings

The equation of motion is a linear fourth-order integro-differential equation.

The approach is exact for single-particle systems and in the high-frequency limit.

The excitation energies satisfy a sum rule ensuring spectral strength accuracy.

Abstract

We derive a closed equation of motion for the current density of an inhomogeneous quantum many-body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integro-differential equation, whose only inputs are the one-particle density matrix and the pair correlation function of the ground-state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle, and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems.