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.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.

Frontiers in PhysicsMar 25, 202018 citations

General Density Matrix Theory

Karl Blum

Springer series on atomic, optical, and plasma physicsOct 4, 20118 citations

Density-matrix theory versus coupled-cluster theory

M. Tohyama

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.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.

Progress of Theoretical and Experimental PhysicsOct 25, 20161 citation

Relation between Density-Matrix Theory and Pairing Theory

M. Tohyama, Satoshi Takahara

The time-dependent density-matrix theory (TDDM) gives a correlated ground state as a stationary solution of the time-dependent equations for one-body and two-body density matrices. The small amplitude limit of TDDM (STDDM) is a version of extended RPA theories which include the effects of ground state correlations. It is shown that the solutions of the Hartree-Fock Bogoliubov theory and the quasi-particle RPA satisfy the TDDM and STDDM equations, respectively, when only pairing-type correlations are taken into account in TDDM and STDDM.The time-dependent density-matrix theory (TDDM) gives a correlated ground state as a stationary solution of the time-dependent equations for one-body and two-body density matrices. The small amplitude limit of TDDM (STDDM) is a version of extended RPA theories which include the effects of ground state correlations. It is shown that the solutions of the Hartree-Fock Bogoliubov theory and the quasi-particle RPA satisfy the TDDM and STDDM equations, respectively, when only pairing-type correlations are taken into account in TDDM and STDDM.

arXiv:nucl-th/0405070Progress of Theoretical PhysicsSep 1, 20044 citations

General Density Matrix Theory

Karl Blum

Springer eBooksJan 1, 19962 citations

^{40}

Ca and

^{48}

Ca and the other the fusion reactions of

^{16}

O +

^{16}