Pairing and surface bosons and the group Sp(4)

Seiya Nishiyama, João da Providência

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 Kahler 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.

Giuseppe E. Santoro, Alessandro Laio, Michele Fabrizio +1 more

Osvaldo Civitarese, M. Reboiro

The Holstein-Primakoff boson mapping technique is applied to transform Hamiltonians defined in terms of the generators of the SU(2) and Sp(4) groups. These solvable models consist of a pair of fermions which interact with hard bosons. It is shown that the mapping to ideal bosons preserves the algebra and that exact matrix elements of the original fermionic Hamiltonian can be reconstructed from the bosonic representation.

Max A. Metlitski, David F. Mross, Subir Sachdev A +1 more

Alfred Wünsche

Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.

P. Van Isacker