Applications of Time-Dependent Density-Matrix Approach

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.

Nuclear ground states in a consistent implementation of the time-dependent density matrix approach

“…When the hierarchy is truncated at some level, the evolution of the correlation tensor C is dictated by an equation which depends on occupation numbers; interaction matrix elements; and the correlation tensor itself. We work under the assumption that genuine three-body correlations are negligible [43,44,52]. In other words, the three-body density matrix is a properly antisymmetrized product of one-body and two-body density matrices only, but does not include any genuine C 3 terms.…”

“…This is presumably challenging, since the BBGKY hierarchy would likely have to be modified. The treatment of genuine three-body correlations may be relevant in nuclear systems too [43]. To break away from the adiabatic evolution picture, one could also attempt to devise an energy minimisation process that included twobody density matrices [79][80][81][82].…”

“…TDDM was first introduced by Cassing and Wang in 1980 [30], and has been applied extensively in nuclear physics by Tohyama [31][32][33][34][35][36][37][38][39][40][41][42][43] and others [44][45][46]. In this context, the TDDM equations are often projected into a moving basis dictated by a TDHF-like equation, plus a time-dependent term that depends on correlations [44].…”

Extended random-phase-approximation study of fragmentation of giant quadrupole resonance in $^{16}$O

“…In the particle-vibration coupling or quasiparticle-phonon models [3] p-h correlations among the 2p-2h configurations are expressed by phonons but the effects of ground-state correlations are neglected. Our extended RPA (ERPA) which is formulated by using the equation-of-motion approach [4] and a correlated ground state in the time-dependent density-matrix theory (TDDM) [5][6][7] includes both the coupling to higher configurations and the effects of ground-state correlations. In ERPA the effects of ground-state correlations are included through the fractional occupation probability n α of a single-particle state α and the correlated part C 2 of a two-body density matrix.…”

Effects of ground-state correlations on damping of giant dipole resonaces in $LS$ closed shell nuclei

“…The particle-vibration coupling or quasiparticle-phonon models [3] express ph correlations included in the 2p-2h configurations by phonons. Our extended RPA (ERPA) derived from the small amplitude limit of the time-dependent densitymatrix theory (TDDM) [4][5][6] consists of the coupled equations for one-body and two-body amplitudes and contains the effects of ground-state correlations through the fractional occupation probability n α of a singleparticle state α and the correlated part C 2 of a two-body density matrix. Special features of ERPA are that the one-body and two-body amplitudes are not restricted to the usual 1p-1h and 2p-2h types.…”