A method of calculating ground-state properties of many particle systems using reduced density matrices

Roman Werpachowski, Jerzy Kijowski

We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.

M. S. L. du Croo de Jongh, J.M Doumen, J. M. J. van Leeuwen

Loren Greenman, David A. Mazziotti

Ground-state two-particle reduced density matrices (2-RDMs) are used to calculate excited-state energy spectra. Solving the Schrodinger equation for excited states dominated by single excitations from the ground-state wavefunction requires the ground-state 2- and 3-RDMs. The excited states, however, can be obtained without a knowledge of the ground-state 3-RDM by two methods: (i) cumulant expansion methods which build the 3-RDM from the 2-RDM, and (ii) double commutator methods which eliminate the 3-RDM. Previous work [Mazziotti, Phys. Rev. A 68, 052501 (2003)] examined the accuracy of excited states extracted from ground-state 2-RDMs, which were calculated by full configuration interaction or the variational 2-RDM method. In this work we employ (i) advances in semidefinite programming to treat the excited states of water and hydrogen fluoride and chains of hydrogen atoms, and (ii) the addition of partial three-particle N-representability conditions to compute more accurate ground-state 2-RDMs. With the hydrogen chains we examine the metal-to-insulator transition as measured by the band gap (the difference between the ground-state and the first excited-state energies), which is difficult for excited-state methods to capture.

Alan Coleman, Vyacheslav I. Yukalov

John M. Herbert, John E. Harriman

We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.