Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0501187 · Physical Review B

Non- V -representability of currents in time-dependent many-particle systems

Roberto D’Agosta, Giovanni Vignale

Published Jun 6, 2005• 26 citations
arXivHTMLPDF

Abstract

We argue that an arbitrarily chosen time-dependent current density is generically non-$V$-representable in a many-particle system; i.e., it cannot be obtained by applying only a time-dependent scalar potential to the system. Furthermore, we show by a concrete example that even a current that is $V$-representable in an interacting many-particle system may (and in general will) turn out to be non-$V$-representable when the interaction between the particles is turned off.

DOI
10.1103/physrevb.71.245103

Related papers

Mapping from current densities to vector potentials in time-dependent current density functional theory

Giovanni Vignale

Under reasonable assumptions the time-dependent particle density $n(\stackrel{P\vec}{r},t)$ and the current density $\stackrel{P\vec}{j}(\stackrel{P\vec}{r},t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\stackrel{P\vec}{r},t)$ and $\stackrel{P\vec}{A}(\stackrel{P\vec}{r},t)$ and is initially in the quantum state $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and starting from an initial state $\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{'}}(0)⟩$, which yields the same density and current as $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$. Here we show that given the initial state of this other many-particle system, the potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.

arXiv:cond-mat/0407682•Physical Review B•Nov 11, 2004•112 citations

On the mapping of time-dependent densities onto potentials in quantum mechanics

Roi Baer

The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.

arXiv:0704.1787•The Journal of Chemical Physics•Jan 24, 2008•45 citations

Comment on ``Time-dependent current-density functional theory for generalized open quantum systems" by J. Yuen-Zhou, C. Rodriiguez-Rosario and A. Aspuru-Guzik

Roberto D’Agosta, Massimiliano Di Ventra

We comment on a recent paper by Yuen-Zhou et al. [ Phys. Chem. Chem. Phys. 2009, 11, 4509 ] which extends some of the results of Time-Dependent Current Density Functional Theory applied to open quantum systems. Besides pointing out some incorrect statements in the theorems, we show that in the proof of the main result of the paper an arbitrary condition is introduced. Moreover, we argue that, since this condition has not a physical origin, one can find two systems with different vector potentials that lead to the same dynamics of the physical quantities of interest, the current and particle densities.

arXiv:0911.3932•Nov 19, 2009•2 citations

V-Representability Beyond Singular Times in Time-Dependent Density Functional Theory: The Hubbard Dimer Case

Éric Cancès, Théo Duez

We study the global $V$-representability problem in time-dependent density functional theory (TDDFT) for the two-particle Hubbard dimer model in the spin-singlet sector. In particular, we investigate whether the time-dependent density of the interacting system obtained for a given one-body external potential $u$ (the tunable control variable) can be reproduced by a non-interacting Kohn--Sham system beyond the singular times at which the usual inversion formulas break down. We deal with both the conventional formulation of TDDFT based on some time-dependent variational principle and the recently introduced geometric formulation of TDDFT (Phys. Rev. Lett. 136 (2026) 256401, Phys. Rev. A 113 (2026) 062222). For both principles, we prove global-in-time $V$-representability, show that singular times induce non-uniqueness of the time-dependent effective potential, and classify the set of solutions.

math-phCross-listedarXiv:2610.33837•Oct 27, 2026•0 citations

Density-functional theory of many-electron systems subjected to time-dependent electric and magnetic fields

Swapan Kumar Ghosh, Asish Kumar Dhara

A time-dependent density-functional formalism is developed for many-electron systems subjected to external electric and magnetic fields with arbitrary time dependence. The single-particle current density is shown to determine uniquely the time-dependent scalar and vector potentials characterizing the system, and hence also the many-particle wave function. A Levy-type universal functional is defined, and practical schemes for the calculation of electron density and the current density through hydrodynamical as well as a set of single-particle Kohn-Sham-like equations are proposed. The gauge invariance of the present self-consistent formalism is also proved.

Physical Review A•Aug 1, 1988•193 citations

Stochastic Time-Dependent Current-Density-Functional Theory

Massimiliano Di Ventra, Roberto D’Agosta

A time-dependent current-density-functional theory for many-particle systems in interaction with arbitrary external baths is developed. We prove that, given the initial quantum state |Psi0> and the particle-bath interaction operator, two external vector potentials A(r,t) and A'(r,t) that produce the same ensemble-averaged current density, j(r,t), must necessarily coincide up to a gauge transformation. This result greatly expands the applicability of time-dependent density-functional theory to open quantum systems, and allows for first-principles calculations of many-particle time evolution beyond Hamiltonian dynamics.

arXiv:cond-mat/0702272•Physical Review Letters•Jun 1, 2007•80 citations
We argue that an arbitrarily chosen time-dependent current density is generically non-VVV-representable in a many-particle system; i.e., it cannot be obtained by applying only a time-dependent scalar potential to the system. Furthermore, we show by a concrete example that even a current that is VVV-representable in an interacting many-particle system may (and in general will) turn out to be non-VVV-representable when the interaction between the particles is turned off.
Under reasonable assumptions the time-dependent particle density $n(\stackrel{P\vec}{r},t)$ and the current density $\stackrel{P\vec}{j}(\stackrel{P\vec}{r},t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\stackrel{P\vec}{r},t)$ and $\stackrel{P\vec}{A}(\stackrel{P\vec}{r},t)$ and is initially in the quantum state $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and starting from an initial state $\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{'}}(0)⟩$, which yields the same density and current as $\ensuremath{\mid}\ensuremath{\psi}(0)⟩$. Here we show that given the initial state of this other many-particle system, the potentials ${V}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$ and ${\stackrel{P\vec}{A}}^{\ensuremath{'}}(\stackrel{P\vec}{r},t)$, if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.
The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.
We comment on a recent paper by Yuen-Zhou et al. [ Phys. Chem. Chem. Phys. 2009, 11, 4509 ] which extends some of the results of Time-Dependent Current Density Functional Theory applied to open quantum systems. Besides pointing out some incorrect statements in the theorems, we show that in the proof of the main result of the paper an arbitrary condition is introduced. Moreover, we argue that, since this condition has not a physical origin, one can find two systems with different vector potentials that lead to the same dynamics of the physical quantities of interest, the current and particle densities.
We study the global VVV-representability problem in time-dependent density functional theory (TDDFT) for the two-particle Hubbard dimer model in the spin-singlet sector. In particular, we investigate whether the time-dependent density of the interacting system obtained for a given one-body external potential uuu (the tunable control variable) can be reproduced by a non-interacting Kohn--Sham system beyond the singular times at which the usual inversion formulas break down. We deal with both the conventional formulation of TDDFT based on some time-dependent variational principle and the recently introduced geometric formulation of TDDFT (Phys. Rev. Lett. 136 (2026) 256401, Phys. Rev. A 113 (2026) 062222). For both principles, we prove global-in-time VVV-representability, show that singular times induce non-uniqueness of the time-dependent effective potential, and classify the set of solutions.
A time-dependent density-functional formalism is developed for many-electron systems subjected to external electric and magnetic fields with arbitrary time dependence. The single-particle current density is shown to determine uniquely the time-dependent scalar and vector potentials characterizing the system, and hence also the many-particle wave function. A Levy-type universal functional is defined, and practical schemes for the calculation of electron density and the current density through hydrodynamical as well as a set of single-particle Kohn-Sham-like equations are proposed. The gauge invariance of the present self-consistent formalism is also proved.
A time-dependent current-density-functional theory for many-particle systems in interaction with arbitrary external baths is developed. We prove that, given the initial quantum state |Psi0> and the particle-bath interaction operator, two external vector potentials A(r,t) and A'(r,t) that produce the same ensemble-averaged current density, j(r,t), must necessarily coincide up to a gauge transformation. This result greatly expands the applicability of time-dependent density-functional theory to open quantum systems, and allows for first-principles calculations of many-particle time evolution beyond Hamiltonian dynamics.