Variational two-particle density matrix calculation for the Hubbard model below half filling using spin-adapted lifting conditions

TL;DR

This paper demonstrates that two-index density matrix methods, enhanced with spin-adapted lifting conditions, can accurately capture the physics of the Hubbard model below half filling, especially in the strong-repulsion regime.

Contribution

It introduces spin-adapted lifting conditions to improve two-particle density matrix calculations for the Hubbard model below half filling, enabling accurate results without heavy three-index conditions.

Findings

Two-index approach fails below half filling without additional conditions.

Lifting conditions improve accuracy in the strong-repulsion limit.

Spin-adapted lifting conditions increase accuracy in intermediate regimes.

Abstract

The variational determination of the two-particle density matrix is an interesting, but not yet fully explored technique that allows 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 {\it et al.}, Phys. Rev. A 73, 062505 (2006)], while a more recent study explored so-called subsystem constraints [N. Shenvi {\it 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 {\it 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.