We describe the response of superfluid nuclei to any external time-dependent probe within an extension of the density functional theory, the time-dependent superfluid local density approximation (TD-SLDA). All quasi-particle wave functions (qpwfs) are represented on a three-dimensional spatial grid to allow for the breaking of all possible symmetries, and subsequently these functions are evolved in time using a high order multi-step algorithm. Mathematically this problem is a system of coupled, time-dependent, nonlinear partial differential equations, which when discretized becomes equivalent to a nonlinear classical mechanics problem with α N x 6 ≈ 10 12 degrees of freedom, where N x is number of spatial mesh points in one direction. The lattice representation of the qpwfs allows for fast and accurate evaluation of various spatial derivatives required for the construction of various physical quantities that appear in the theory. The total number of independent qpwfs is of the order of spatial mesh points, and the system is evolved for tens to hundreds of thousands of time steps. We describe how we implemented the TD-SLDA method and report some of the challenges encountered.
We describe the response of superfluid nuclei to any external time-dependent probe within an extension of the density functional theory, the time-dependent superfluid local density approximation (TD-SLDA). All quasi-particle wave functions (qpwfs) are represented on a three-dimensional spatial grid to allow for the breaking of all possible symmetries, and subsequently these functions are evolved in time using a high order multi-step algorithm. Mathematically this problem is a system of coupled, time-dependent, nonlinear partial differential equations, which when discretized becomes equivalent to a nonlinear classical mechanics problem with α N x 6 ≈ 10 12 degrees of freedom, where N x is number of spatial mesh points in one direction. The lattice representation of the qpwfs allows for fast and accurate evaluation of various spatial derivatives required for the construction of various physical quantities that appear in the theory. The total number of independent qpwfs is of the order of spatial mesh points, and the system is evolved for tens to hundreds of thousands of time steps. We describe how we implemented the TD-SLDA method and report some of the challenges encountered.
The time-dependent superfluid local density approximation (TDSLDA) is an extension of the Hohenberg-Kohn density functional theory (DFT) to time-dependent phenomena in superfluid fermionic systems. Unlike linear response theory, which is only valid for weak external fields, the (TDSLDA) approach allows one to study non-linear excitations in fermionic superfluids, including large amplitude collective modes, and the response to strong external probes. Even in the case of weak external fields, the (TDSLDA) approach is technically easier to implement. We will illustrate the implementation of the (TDSLDA) for the unitary Fermi gas, where dimensional arguments and Galilean invariance simplify the form of the functional, and ab initio input from (QMC) simulations fix the coefficients to quite high precision.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
Time-dependent density-functional theory is mathematically formulated through nonlinear coupled time-dependent three-dimensional partial differential equations, and it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schr\"odinger equation for an interacting many-body system is, however, linear and mathematically the exact equations of the density-functional theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a density-functional theory framework. Whether for realistic implementations of the time-dependent density-functional theory the question of the absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density-functional theory is a unique tool allowing us to study the nonequilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Our analysis, for a number of quantum superfluid many-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom O(1010) and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.
The first detailed comparison between ab initio calculations of finite fermionic superfluid systems, performed recently by Chang and Bertsch [Phys. Rev. A 76, 021603(R) (2007)] and by von Stecher, Grange, and Blume [e-print arXiv:0705.0671v1] and the extension of the density-functional theory superfluid local-density approximation (SLDA) is presented. It is shown that SLDA reproduces the total energies, number density distributions in inhomogeneous systems along with the energy of the normal state in homogeneous systems. Unlike the Kohn-Sham LDA, in SLDA the effective fermion mass differs from the bare fermion mass and the spectrum of elementary excitations is also reproduced.
Time-dependent density functional theory (TDDFT) for quantum many-body systems on a lattice is formulated rigorously. We prove the uniqueness of the density-to-potential mapping and demonstrate that a given density is v representable if the initial many-body state and the density satisfy certain well-defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is locally in time v representable and therefore the lattice TDDFT is guaranteed to exist. The TDDFT existence and uniqueness theorem is valid for any connected lattice, independently of its size, geometry, and/or spatial dimensionality. General statements of the existence theorem are illustrated on a pedagogical exactly solvable example, which displays all the details and subtleties of the proof in a transparent form. In conclusion we briefly discuss remaining open problems and directions for future research.
Abstract. Using the time-dependent theory of quantum mechanics, we investigate nuclear electric dipole responses. The time evolution of a wave function is explicitly calculated in the coordinatespace representation. The particle continuum is treated with the absorbing boundary condition. Calculated time-dependent quantities are transformed into those of familiar energy representation. We apply the method to a three-body model for 11 Li and to the mean-field model for 22 O, then discuss properties of E1 response. TIME-DEPENDENT METHOD FOR NUCLEAR RESPONSES Atomic nuclei exhibit a variety of responses to different experimental probes; Coulomb and nuclear excitations, spin- and isospin-dependent probes, high- and low-energy reactions. In order to investigate properties of these nuclear excitations, it is useful to study response function for a probe (external) operator of interest. Normally, these quantum many-body problems are theoretically studied using the energy representation, in which calculations are carried out by either diagonalizing the Hamiltonian matrices or solving differential equations with a fixed energy. These approaches are advantageous when we are interested in states within a limited energy range. However, when we calculate