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Frontiers in Physics

Applications of Time-Dependent Density-Matrix Approach

M. Tohyama

Published Mar 25, 2020• 18 citations
Publisher

Abstract

The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of $^{40}$Ca and $^{48}$Ca and the other the fusion reactions of $^{16}$O + $^{16}$O.

DOI
10.3389/fphy.2020.00067

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The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of 40^{40}40Ca and 48^{48}48Ca and the other the fusion reactions of 16^{16}16O + 16^{16}16O.
The Bogoliubov-Born-Green-Kirkwood-Yvon or time-dependent density matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three-body level in approximating the three-body correlation function by a quadratic form of two-body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in nontrivial model cases that the approximate inclusion of three-body correlation functions is very important to obtain precise results. A small amplitude approximation of this time-dependent nonlinear equation for the two-body correlation function is performed (STDDM*-b) and it is shown that the one-body sector of this generalized nonlinear second random phase approximation (RPA) equation is equivalent to the self-consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three-body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.
The ground state of 16O^{16}\mathrm{O}16O is calculated by using a time-dependent density-matrix approach derived from a new truncation scheme of the Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by an antisymmetrized product of two-body density matrices. The new scheme is compared with a simpler truncation scheme previously used for the calculation of the ground state of 16O^{16}\mathrm{O}16O where the three-body density matrix is neglected and only two-particle--two-hole elements of the two-body density matrix are considered. It is shown that the results obtained from the two truncation schemes agree well with the exact solution.
Time-dependent density functional theory (TDDFT) is widely used for understanding and predicting properties and behaviors of matter. As one of the fundamental theorems in TDDFT, Van Leeuwen theorem [Phys. Rev. Lett. 82, 3863 (1999)PRLTAO0031-900710.1103/PhysRevLett.82.3863] guarantees how to construct a unique potential with the same one-body density evolution. Here we extend Van Leeuwen theorem by exploring truncation criteria in Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Our generalized theorem demonstrates the existence of a unique nonlocal potential to accurately reconstruct the multibody density evolution in binary interacting systems. Under nonstringent conditions, truncation of the BBGKY-hierarchy equations aligns with the behavior of multibody density evolution and maintains consistency in the reduced equations. Based on this theorem, we further construct a static many-body potential and extend the well-known Casida equation for multiple-excitation energy. The extended TDDFT provides an accurate and first-principles framework capable of describing the kinetic processes of correlated systems, including strongly coupled particle transport, multiple-excitation, and ionization processes.
In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.
Coupled-cluster doubles theory (CCD) consists of the equation for two-particle–two-hole amplitudes. Time-dependent density-matrix theory (TDDM) consists of the equations of motion for one-body and two-body density matrices. Similarities and differences between CCD and TDDM are investigated. They are also applied to model Hamiltonians, and a comparison with the results of an exact diagonalization approach is made. It is pointed out that the effect of the fractional occupation of the single-particle states, which is included in TDDM, plays a role in suppressing two-body correlations in strongly interacting regimes.
Applications of an extended version of the Hartree-Fock theory and the random-phase approximation derived from the time-dependent density-matrix theory (TDDM) are presented. In this TDDM-based theory, the ground state is given as a stationary solution of the TDDM equations and the excited states are calculated using the small-amplitude limit of TDDM. The first application presented is an extended Lipkin model in which an interaction term describing a particle scattering is added to the original Hamiltonian so that the damping of a collective state is taken into account. It is found that the TDDM-based theory well reproduces the ground state and excited states of the extended Lipkin model. The quadrupole excitation of the oxygen isotopes 16,20,22O^{16,20,22}\mathrm{O}16,20,22O is also studied as realistic applications of the TDDM-based theory. It is found that large fragmentation of the giant quadrupole resonance in 16O^{16}\mathrm{O}16O is reproduced, and it is pointed out that the effects of ground-state correlations are quite important for fragmentation. It is also found that the quadrupole states in neutron-rich oxygen isotopes have small spreading widths.