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:1001.0616 · Physical Review B

Continuum mechanics for quantum many-body systems: Linear response regime

Gao Xianlong, Jianmin Tao, Giovanni Vignale, I. V. Tokatly

Published May 7, 2010• 32 citations
arXivHTMLPDF

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 integrodifferential 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.

DOI
10.1103/physrevb.81.195106

Related papers

Linear Continuum Mechanics for Quantum Many-Body Systems

Jianmin Tao, Gao Xianlong, Giovanni Vignale +1 more

We develop the continuum mechanics of quantum many-body systems in the linear response regime. The basic variable of the theory is the displacement field, for which we derive a closed equation of motion under the assumption that the time-dependent wave function in a locally comoving reference frame can be described as a geometric deformation of the ground-state wave function. We show that this equation of motion is exact for systems consisting of a single particle, and for all systems at sufficiently high frequency, and that it leads to an excitation spectrum that has the correct integrated strength. The theory is illustrated by simple model applications to one- and two-electron systems.

arXiv:0904.4783•Physical Review Letters•Aug 18, 2009•23 citations

Time-dependent deformation functional theory

I. V. Tokatly

We present a constructive derivation of a time-dependent deformation functional theory---a collective variable approach to the nonequilibrium 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 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 construction of nonadiabatic approximations. We also discuss a relation of the present approach to a recent continuum mechanics of the incompressible quantum Hall liquids.

arXiv:cond-mat/0612254•Physical Review B•Mar 5, 2007•49 citations

Dynamic linear response of many-electron systems: An integral formulation of density-functional theory

Weitao Yang

A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.

Physical Review A•Dec 1, 1988•22 citations

Time-Dependent Quantum Many-Body Systems: Linear Response, Electronic Transport, and Reduced Density Matrices

Heiko Appel

Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.

•Jan 1, 2007•10 citations

Many-body theory for systems with particle conversion: Extending the multiconfigurational time-dependent Hartree method

Ofir E. Alon, Alexej I. Streltsov, Lorenz S. Cederbaum

We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.

arXiv:0811.3853•Physical Review A•Feb 4, 2009•43 citations

Linear response theory revisited III: One-body response formulas and generalized Boltzmann equations

M. Charbonneau, K. M. van Vliet, Pagiotis Vasilopoulos

The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.

Journal of Mathematical Physics•Feb 1, 1982•162 citations
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 integrodifferential 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.
We develop the continuum mechanics of quantum many-body systems in the linear response regime. The basic variable of the theory is the displacement field, for which we derive a closed equation of motion under the assumption that the time-dependent wave function in a locally comoving reference frame can be described as a geometric deformation of the ground-state wave function. We show that this equation of motion is exact for systems consisting of a single particle, and for all systems at sufficiently high frequency, and that it leads to an excitation spectrum that has the correct integrated strength. The theory is illustrated by simple model applications to one- and two-electron systems.
We present a constructive derivation of a time-dependent deformation functional theory---a collective variable approach to the nonequilibrium 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 Green's deformation tensor gij{g}_{ij}gij​. 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 construction of nonadiabatic approximations. We also discuss a relation of the present approach to a recent continuum mechanics of the incompressible quantum Hall liquids.
A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.
Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.
We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.