Variational reduced-density-matrix calculation of the one-dimensional Hubbard model

Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an $N$-particle wave function by two sets of $N$-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by $N$-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.

Figure 1

The total, connected, and unconnected energies as functions of $U$ for the 6-site half-filled Hubbard model. The connected (correlation) energy grows dramatically after $U=6$, while the total energy remains fairly constant due to significant cancelation of the connected and unconnected energies. Energies are computed from the connected part of the 2-RDM obtained from FCI.