arXiv subjects

Joao Da Providencia

Publications and source records attributed to Joao Da Providencia.

3 recordsLinked to original sources

arXiv2019

Mean-field theory based on the \mathfrak{Jacobi~hsp} := semi-direct sum \mathfrak{h}_N \rtimes \mathfrak{sp}(2N,\mathbb{R})_\mathbb{C} algebra of boson operators

In this paper, we give an expression for canonical transformation group with Grassmann variables, basing on the \mathfrak{Jacobi~hsp} \!:= semi-direct sum \mathfrak{h}_{N} \rtimes \mathfrak{sp}(2N,\mathbb{R})_\mathbb{C} algebra of boson operators. We assume a mean-field Hamiltonian (MFH) linear in the \mathfrak{Jacobi} generators. We diagonalize the boson MFH. We show a new aspect of eigenvalues of the MFH. An excitation energy arisen from additional self-consistent field (SCF) parameters has never been seen in the traditional boson MFT. We derive this excitation energy. We extend the Killing potential in the \frac{Sp(2N,\mathbb{R})_\mathbb{C}}{U(N)} coset space to the one in the \frac{Sp(2N+2,\mathbb{R})_\mathbb{C}}{U(N+1)} coset space and make clear the geometrical structure of Kähler manifold, a non-compact symmetric space \frac{Sp(2N+2,\mathbb{R})_\mathbb{C}}{U(N+1)}. The \mathfrak{Jacobi~hsp} transformation group is embedded into an Sp(2N+2,\mathbb{R})_\mathbb{C} group and an \frac{Sp(2N+2,\mathbb{R})_\mathbb{C}}{U(N+1)} coset variable is introduced. Under such mathematical manipulations, extended bosonization of Sp(2N+2,\mathbb{R})_\mathbb{C} Lie operators, vacuum function and differential forms for extended boson are presented by using integral representation of boson state on the \frac{Sp(2N+2,\mathbb{R})_\mathbb{C}}{U(N+1)} coset variables.

hep-th↗

arXiv2019

Remarks on the Mean-Field Theory Based on the SO(2N+1) Lie Algebra of the Fermion Operators

Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing paired- and 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 Kaehler 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.

hep-th↗

arXiv2012

A new description of motion of the Fermionic SO(2N+2) top in the classical limit under the quasi-anticommutation relation approximation

The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidian space (N: number of single-particle states of the fermions). The images of fermion annihilation-creation operators must satisfy the canonical anti-commutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N+1) self-consistent field (SCF) Hartree-Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle-hole plus BCS type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation, due to the unpaired-mode effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi anti-commutation-relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters k and l in the Lagrange multipliers can be determined withuout any inconcistency.

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