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

Abstract

The variational determination of the two-particle density matrix is an interesting, but not yet fully explored technique that allows us to obtain ground-state properties of a quantum many-body system without reference to an N-particle wave function. The one-dimensional fermionic Hubbard model has been studied before with this method, using standard two- and three-index conditions on the density matrix [J. R. Hammond et al., Phys. Rev. A 73, 062505 (2006)], while a more recent study explored so-called subsystem constraints [N. Shenvi et al., Phys. Rev. Lett. 105, 213003 (2010)]. These studies reported good results even with only standard two-index conditions, but have always been limited to the half-filled lattice. In this Letter, we establish the fact that the two-index approach fails for other fillings. In this case, a subset of three-index conditions is absolutely needed to describe the correct physics in the strong-repulsion limit. We show that applying lifting conditions [J. R. Hammond et al., Phys. Rev. A 71, 062503 (2005)] is the most economical way to achieve this, while still avoiding the computationally much heavier three-index conditions. A further extension to spin-adapted lifting conditions leads to increased accuracy in the intermediate repulsion regime. At the same time, we establish the feasibility of such studies to the more complicated phase diagram in two-dimensional Hubbard models.

The second-order reduced density matrix method and the two-dimensional Hubbard model

“…This model is interesting not only because of its simplicity, but also its capability of describing strong electron correlation. The RDM method has been applied to the Hubbard model by Hammond et al [12], Nakata et al [9], and Verstichel et al [13]. Their results very accurately described total energies as well as other properties.…”

Structure of the one‐electron reduced density matrix from the generalized Pauli exclusion principle

“…The definition of the set of pure N ‐representable ‐RDMs by the generalized Pauli conditions may provide new insights into the development of practical 1‐RDM‐based electronic structure methods . The generalized Pauli conditions may also be useful in some cases as further restrictions on the N ‐representability of the 2‐RDM in variational calculations based on the 2‐RDM . As suggested by other authors, pinning of the 1‐RDM to the generalized Pauli conditions provides potentially useful information about the Slater determinants that contribute most significantly to the wave function.…”

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

“…The Hubbard model has been studied before using the v2DM method, see e.g. [19,[33][34][35], but only the energy was considered, and this for relatively small lattice sizes (up to L = 14). 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.…”

“…To obtain a decent accuracy, however, it will be necessary to include the three-index constraints, which is computationally hard. One way around this was set forward in [35] with the use of lifting conditions [16,18]. …”