Non-Abelian density functional theory

Rosensteel, G.; Dankova, Ts.

doi:10.1088/0305-4470/31/44/017

Cited by 10 publications

(4 citation statements)

References 13 publications

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“…Further we point out that the present MFT is deeply related to the algebraic MFT (AMFT) which has been proposed by Rosensteel and Dancova [29,30,31] based on the coadjoint orbit method [32]. Following them, when we try to apply the AMFT to the present theory, every time it is indispensably necessary to introduce an even-dimensional GDM on the SO(2N+2) CS rep as is seen soon later.…”

Section: Discussion Perspective and Summarymentioning

confidence: 74%

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Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Nishiyama

Providência

2019

Int. J. Geom. Methods Mod. Phys.

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Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing pairedand unpaired-mode effects, in this paper, we propose a generalized HB (GHB) MFH in terms of the SO(2N+1) Lie algebra of fermion pair and creation-annihilation operators. We diagonalize the GHB-MFH and throughout the diagonalization of which, we can first obtain the unpaired mode amplitudes which are given by the SCF parameters appeared in the HBT together with the additional SCF parameter in the GHB-MFH and by the parameter specifying the property of the SO(2N +1) group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such the results. We construct the Killing potential in the coset space SO(2N ) U (N ) on the Kähler symmetric space which is equivalent with the generalized density matrix (GDM). We show another approach to the fermion MFH based on such a GDM. We derive an SO(2N+1) GHB MF operator and a modified HB eigenvalue equation. We discuss on the MF theory related to the algebraic MF theory based on the GDM and the coadjoint orbit leading to the non-degenerate symplectic form.

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Section: Discussion Perspective and Summarymentioning

confidence: 74%

“…There exists an alternative way, the so-called "moment map "method proposed by Guillemin and Sternberg [36] and by Ref. [31]. This method may be also considered to be a powerful tool for constructing the microscopic theory to describe motions with strongly collective correlations.…”

Section: Discussion Perspective and Summarymentioning

confidence: 99%

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Nishiyama

Providência

2019

Int. J. Geom. Methods Mod. Phys.

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“…Density functional theory is concerned with discovering accurate energy functionals, primarily for applications to chemistry. A generalization of the Hohenberg-Kohn theorem guarantees the existence of an energy functional whose minimum is the exact ground state g-density [40]. Although this theorem motivates the search for the holy grail energy functional, researchers are satisfied usually with a useful approximation to it.…”

Section: Discussionmentioning

confidence: 99%

“…When d=6, m=4, Q=2, α=0.333, and N 0 =1/2, Levenberg-Marquardt minimization yields the following matrices: Note that the 5×5 block submatrices of the d=6 pentadiagonal matrices are identical to the d=5 tridiagonal calculation of equation (40). The calculated matrices are tridiagonal; a zero on the Q=2 pentadiagonals indicates that the absolute value of this calculated matrix entry is smaller than 10 −7 .…”

Section: Pentadiagonal Ansatzmentioning

confidence: 99%

Equations of motion for noncompact symplectic dynamical symmetry

Whitaker¹,

Rosensteel²

2019

Phys. Scr.

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When the high dimension of the quantum Hilbert space creates a serious challenge for conventional numerical diagonalisation of the Hamiltonian matrix, the equations of motion (EOM) method may solve efficiently the eigenvalue problem in some instances. EOM assumes the existence of a spectrum generating Lie algebra (SGA) for which the Hamiltonian operator is a polynomial (typically a quadratic) in the algebra’s basis operators. The method computes values of low-energy excitation energies and corresponding algebra matrix elements by solving a coupled set of commutation relations. To test EOM’s effectiveness, the paper applies it to SU(1, 1) Sp(1, ), whose unitary discrete series representation spaces are infinite dimensional. The paper investigates two (1, 1) Hamiltonians. The first Hamiltonian has a phase transition that challenges brute force diagonalisation. The second Hamiltonian is the one-dimensional quartic potential. For the quartic case, the algebraic mean field Hamiltonian, an approximation to the SGA Hamiltonian, provides a qualitative solution and a useful starting point for an EOM computation. EOM yields fast and accurate results on a laptop computer for both Hamiltonians.

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SU(3) mean field Hamiltonian

Rosensteel

Dankova

2002

J. Phys. A: Math. Gen.

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Non-Abelian density functional theory

Cited by 10 publications

References 13 publications

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Equations of motion for noncompact symplectic dynamical symmetry

SU(3) mean field Hamiltonian

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