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Variational density study of the one-dimensional Hubbard model: the need for three-particle constraints

Brecht Verstichel, Helen van Aggelen, Ward Poelmans, Dimitri Van Neck

Published Jan 1, 2012• 0 citations
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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.

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Variational determination of the two-particle density matrix as a quantum many-body technique

Brecht Verstichel

In this thesis the variational optimisation of the density matrix is discussed as a method in many-body quantum mechanics. This is a relatively unknown technique in which one tries to obtain the two-particle reduced density matrix directly in a variational approach. In the first Chapter we introduce the subject in the broader context of many-body quantum mechanics, and briefly sketch the history of the field. The second Chapter tries to summarise what is known about N-representability of reduced density matrices, and derives these results in a unified framework. The optimisation problem can be formulated as a semidefinite program. Three different algorithms are discussed in Chapter three, and their performance is compared. In Chapter four we show how symmetry can be exploited to reduce the computational cost considerably. Several applications of the method are discussed in Chapter five. We show that the standard two-index constraints fail in the strong-correlation limit of the one-dimensional Hubbard model. In Chapter six we identify the spin-adapted lifting conditions as the most compact constraints that can correctly describe this limit.

arXiv:1203.5659•Mar 26, 2012•4 citations
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.
In this thesis the variational optimisation of the density matrix is discussed as a method in many-body quantum mechanics. This is a relatively unknown technique in which one tries to obtain the two-particle reduced density matrix directly in a variational approach. In the first Chapter we introduce the subject in the broader context of many-body quantum mechanics, and briefly sketch the history of the field. The second Chapter tries to summarise what is known about N-representability of reduced density matrices, and derives these results in a unified framework. The optimisation problem can be formulated as a semidefinite program. Three different algorithms are discussed in Chapter three, and their performance is compared. In Chapter four we show how symmetry can be exploited to reduce the computational cost considerably. Several applications of the method are discussed in Chapter five. We show that the standard two-index constraints fail in the strong-correlation limit of the one-dimensional Hubbard model. In Chapter six we identify the spin-adapted lifting conditions as the most compact constraints that can correctly describe this limit.