Time-dependent deformation functional theory

TL;DR

This paper develops a time-dependent deformation functional theory for quantum many-body systems, linking fluid element motion to a universal stress functional of the deformation tensor, and connects it to current density functional theory.

Contribution

It introduces a novel collective variable approach based on deformation tensors, providing a universal functional framework for nonequilibrium quantum dynamics.

Findings

Derivation of a hydrodynamics equation governed by a universal stress functional.

Representation of the theory in both hydrodynamic and Kohn-Sham formulations.

Exact forms of exchange-correlation potentials and implications for nonadiabatic approximations.

Abstract

We present a constructive derivation of a time-dependent deformation functional theory -- a collective variable approach to the nonequalibrium quantum many-body problem. It is shown that the motion of infinitesimal fluid elements (i.e. a set of Lagrangian trajectories) in an interacting quantum system is governed by a closed hydrodynamics equation with the stress force being a universal functional of the Green's deformation tensor $g_{ij}$. Since the Lagrangian trajectories uniquely determine the current density, this approach can be also viewed as a representation of the time-dependent current density functional theory. To derive the above theory we separate a "convective" and a "relative" motions of particles by reformulating the many-body problem in a comoving Lagrangian frame. Then we prove that a properly defined many-body wave function (and thus any observable) in the comoving frame is a universal functional of the deformation tensor. Both the hydrodynamic and the Kohn-Sham formulations of the theory are presented. In the Kohn-Sham formulation we derive a few exact representations of the exchange-correlation potentials, and discuss their implication for the construction of new nonadiabatic approximations. We also discuss a relation of the present approach to a recent continuum mechanics of the incompressible quantum Hall liquids.