A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

Abstract

The quantum many-body problem can be rephrased as a variational determination of the twobody reduced density matrix, subject to a set of N -representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N -representability conditions perform very well for this problem. * brecht.verstichel@ugent.be

Benchmarking the Variational Reduced Density Matrix Theory in the Doubly Occupied Configuration Interaction Space with Integrable Pairing Models

“…The optimization thus finds a lower bound to the exact ground-state energy and an approximation to the exact ground-state 2RDM. Such an approach, known as the variational second-order reduced density matrix (v2RDM) method has been applied with different degrees of success in quantum-chemistry problems, [24][25][26][27] nuclear-physics, 28,29 and condensed-matter. [30][31][32] Recently, the computational efficiency of the v2RDM method has been substantially improved for systems whose states can be accurately described in terms of doubly-occupied single-particle states only.…”

Variational Two-Particle Density Matrix Calculation for the Hubbard Model Below Half Filling Using Spin-Adapted Lifting Conditions

“…The other five conditions are spin-adapted generalizations of the lifting conditions introduced in [1,13,23], and are of the form: (18) in which the B † consist of different combinations of creation and annihilation operators. As an example of such a condition, consider B † defined as: The numerical optimization of the 2.5DM under these positivity constraints is a semidefinite program, and exactly the same methods used for the optimization of the 2DM can be used [4,24,25]. The scaling of the basic matrix manipulations in this optimization is M 7 , as opposed to the full three-index conditions, which scale as M 9 , with M the size of single-particle Hilbert space.…”

Extensive v2DM study of the one-dimensional Hubbard model for large lattice sizes: Exploiting translational invariance and parity

“…In this paper we study different filling factors, and extract various properties like the ground-state energy and two-particle correlation functions in order to assess the quality of the variationally obtained 2DM. The v2DM results discussed in this Section were all obtained using the primal-dual predictor corrector semidefinite programming algorithm [28]. Although the one-dimensional Hubbard model can be solved exactly using the Bethe ansatz [36][37][38][39], it is hard to extract information about the solution for finite systems.…”

“…We have implemented three different semidefinite programming algorithms. Two are so-called interior point methods (a dual-only potential reduction algorithm, [32] and a primal-dual interior point algorithm [28]), where the 2DM is optimized from within the N -representable region. In the third one (a boundary point method [27]) the 2DM is not required to be N -representable during the optimization.…”

“…A lot of activity has been devoted to the search for new N -representability conditions, which improve the result in a computationally cheap way [24,25]. There have also been efforts to improve the semidefinite programming algorithms by adapting them to the specific problem of density matrix optimization [26][27][28], allowing the study larger systems.…”