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arXiv:cond-mat/0410205 · Physical Review B

Quantum many-body dynamics in a Lagrangian frame: II. Geometric formulation of time-dependent density functional theory

I. V. Tokatly

Published Apr 4, 2005• 75 citations
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Abstract

We formulate equations of time-dependent density functional theory (TDDFT) in the comoving Lagrangian reference frame. The main advantage of the Lagrangian description of many-body dynamics is that in the comoving frame the current density vanishes, while the density of particles becomes independent of time. Therefore a comoving observer will see a picture which is very similar to that seen in the equilibrium system from the laboratory frame. It is shown that the most natural set of basic variables in TDDFT includes the Lagrangian coordinate, $\mathbf{\ensuremath{\xi}}$, a symmetric deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, and a skew-symmetric vorticity tensor, ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$. These three quantities, respectively, describe the translation, deformation, and the rotation of an infinitesimal fluid element. Reformulation of TDDFT in terms of new basic variables resolves the problem of nonlocality and thus allows us to regularly derive a local nonadiabatic approximation for exchange correlation (xc) potential. Stationarity of the density in the comoving frame makes the derivation to a large extent similar to the derivation of the standard static local density approximation. We present a few explicit examples of nonlinear nonadiabatic xc functionals in a form convenient for practical applications.

DOI
10.1103/physrevb.71.165105

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We formulate equations of time-dependent density functional theory (TDDFT) in the comoving Lagrangian reference frame. The main advantage of the Lagrangian description of many-body dynamics is that in the comoving frame the current density vanishes, while the density of particles becomes independent of time. Therefore a comoving observer will see a picture which is very similar to that seen in the equilibrium system from the laboratory frame. It is shown that the most natural set of basic variables in TDDFT includes the Lagrangian coordinate, \ensuremathξ\mathbf{\ensuremath{\xi}}\ensuremathξ, a symmetric deformation tensor g\ensuremathμ\ensuremathν{g}_{\ensuremath{\mu}\ensuremath{\nu}}g\ensuremathμ\ensuremathν​, and a skew-symmetric vorticity tensor, F\ensuremathμ\ensuremathν{F}_{\ensuremath{\mu}\ensuremath{\nu}}F\ensuremathμ\ensuremathν​. These three quantities, respectively, describe the translation, deformation, and the rotation of an infinitesimal fluid element. Reformulation of TDDFT in terms of new basic variables resolves the problem of nonlocality and thus allows us to regularly derive a local nonadiabatic approximation for exchange correlation (xc) potential. Stationarity of the density in the comoving frame makes the derivation to a large extent similar to the derivation of the standard static local density approximation. We present a few explicit examples of nonlinear nonadiabatic xc functionals in a form convenient for practical applications.
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.
We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ and a skew-symmetric vorticity tensor ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$, where $\mathbf{\ensuremath{\xi}}$ in the Lagrangian coordinate. Equations of motion are equivalent to those for a quantum many-body system in a space with time-dependent metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$ in the presence of an effective magnetic field ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathbf{\ensuremath{\xi}},t)$. To illustrate the general formalism we apply it to the proof of the harmonic potential theorem. As another example of application we consider a fast long wavelength dynamics of a Fermi system in the dynamic Hartree approximation. In this case the kinetic equation in the Lagrangian frame can be solved explicitly. This allows us to formulate the description of a Fermi gas in terms of an effective nonlinear elasticity theory. We also discuss a relation of our results to time-dependent density functional theory.
Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential, Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t), defined as a nonlocal functional of the density $n({\mathbf{r}}^{\ensuremath{'}},{t}^{\ensuremath{'}})$. The popular adiabatic local-density approximation (ALDA) for Vxc(r,t){V}_{\mathrm{xc}}(\mathbf{r},t)Vxc​(r,t) uses only densities at the same space-time point (r,t)(\mathbf{r},t)(r,t). To go beyond the ALDA, two local approximations have been proposed based on quantum hydrodynamics and elasticity theory: (a) using the current as the basic variable (C-TDDFT) [G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4847 (1997)], (b) working in a comoving Lagrangian reference frame (L-TDDFT) [I. V. Tokatly, Phys. Rev. B 71, 165105 (2005)]. In this paper we illustrate, compare, and analyze both nonadiabatic theories for simple time-dependent model densities in the linear and nonlinear regime, for a broad range of time and frequency scales. C- and L-TDDFT are identical in certain limits, but, in general, exhibit qualitative and quantitative differences in their respective treatment of elastic and dissipative electron dynamics. In situations where the electronic density rapidly undergoes large deformations, it is found that nonadiabatic effects can become significant, causing the ALDA to break down.
Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.
Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schrödinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schrödinger evolution that has not been previously explored. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest new strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive novel Kohn--Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these new approaches.
The logical structure and basic theorems of time-dependent current density functional theory (TDCDFT) are analyzed and reconsidered from the point of view of recently proposed time-dependent deformation functional theory (TDDefFT). It is shown that the formalism of TDDefFT allows to avoid a traditional external potential-to-density/current mapping. Instead the theory is formulated in a form similar to the constrained search procedure in the ground state DFT. Within this formulation of TDCDFT all basic functionals appear from the solution of a constrained universal many-body problem in a comoving reference frame, which is equivalent to finding a conditional extremum of a certain universal action functional. As a result the physical origin of the universal functionals entering the theory, as well as their proper causal structure become obvious. In particular, this leaves no room for any doubt concerning the predictive power of the theory.