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

Anderson, James S.; Nakata, Masao; Igarashi, Ryo; Yamashita, Makoto

doi:10.1016/j.comptc.2012.08.018

Cited by 24 publications

(37 citation statements)

References 55 publications

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“…For consistency the symmetry then has to be enforced in these blocks by imposing linear constraints on the 2DM during the optimization. [39], and it is shown that they correspond.…”

Section: Point Group Symmetrymentioning

confidence: 91%

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

et al. 2014

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The reduced density matrix is variationally optimized for the two-dimensional Hubbard model. Exploiting all symmetries present in the system, we have been able to study 6×6 lattices at various fillings and different values for the on-site repulsion, using the highly accurate but computationally expensive three-index conditions. To reduce the computational cost we study the performance of imposing the three-index constraints on local clusters of 2×2 and 3×3 sites. We subsequently derive new constraints which extend these cluster constraints to incorporate the open-system nature of a cluster on a larger lattice. The feasibility of implementing these new constraints is demonstrated by performing a proof-of-principle calculation on the 6 × 6 lattice. It is shown that a large portion of the three-index result can be recovered using these extended cluster constraints, at a fraction of the computational cost.

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“…For consistency the symmetry then has to be enforced in these blocks by imposing linear constraints on the 2DM during the optimization. [39], and it is shown that they correspond.…”

Section: Point Group Symmetrymentioning

confidence: 91%

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

et al. 2014

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“…As a consequence, the two less stringent conditions T 1 and T 2 , originally proposed by Erdahl [52], and implemented for small atoms and molecules by Zhao et al [22] are used in practical calculations. They read: (24) which are both positive semidefinite. The fundamental interest of Eqs.…”

Section: B N-representability Conditions and Many-body Quantum Numbersmentioning

confidence: 99%

“…These conditions consist in positive semidefinite operators, whose expectation value thus has to be non-negative when evaluated with a 2RDM. Moreover, they can be systematically arranged in a hierarchy where each level yields an increasingly tighter lower bound on the exact ground-state energy [15,16,18,[23][24][25]. However, limitations in optimization software and computer resources prevented the practical development of the 2RDM method for several years.…”

Section: Introductionmentioning

confidence: 99%

Revisiting the variational two-particle reduced density matrix for nuclear systems

Li,

Michel,

Zuo

et al. 2021

Preprint

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In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.

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“…(1) was unknown until recently, 8 an approximate class of Nrepresentability constraints has been demonstrated to be sufficient for calculating ground-state properties of the metal-to-insulator transition of hydrogen chains, 9 ground states and charge distributions of quantum dots, 10 quantum phase transitions, 11,12 dissociation channels, 13 and quantum lattice systems. [14][15][16][17][18][19][20] Variational minimization of the energy as a functional of the 2-RDM is expressible as a special convex optimization problem known as a semidefinite program.…”

Section: Introductionmentioning

confidence: 99%

“…While recent RDM calculations have examined linear 14 as well as 4 × 4 and 6 × 6 Hubbard lattices, 15,20 there has not been an exploration of RDMs on quasi-one-dimensional Hubbard lattices with a comparison to the one-dimensional Hubbard lattices. How does the electron correlation change as we move from a one-dimensional to a quasi-one-dimensional Hubbard model?…”

mentioning

confidence: 99%

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Rubin¹,

Mazziotti²

2014

Highlights in Theoretical Chemistry

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Minimizing the energy of an N -electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary N -representability conditions (conditions for the 2-RDM to represent an ensemble N -electron quantum system), yields a rigorous lower bound to the groundstate energy in contrast to variational wavefunction methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-one-dimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one-and quasi-one-dimensional lattices for both half filling and less-than-half filling.

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

Cited by 24 publications

References 55 publications

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

Revisiting the variational two-particle reduced density matrix for nuclear systems

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

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