© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1107.5704 · Journal of Physics A Mathematical and Theoretical

Quasibosons composed of twoq-fermions: realization by deformed oscillators

A M Gavrilik, I I Kachurik, Yu A Mishchenko

Published Nov 3, 2011• 41 citations

Abstract

Composite bosons, here called quasibosons (e.g. mesons, excitons, etc), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the 'fermion+fermion' composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q -fermions ( q -deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is a quadratic polynomial in the number operator. In the case of two q -fermions, q ≠ 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.

DOI: 10.1088/1751-8113/44/47/475303

Related papers

Two-fermion composite quasi-bosons and deformed oscillators

Alexandre M. Gavrilik, І. І. Качурик, Yu. A. Mishchenko

The concept of quasi-bosons or composite bosons (like mesons, excitons etc.) has a wide range of potential physical applications. Even composed of two pure fermions, the quasi-boson creation and annihilation operators satisfy non-standard commutation relations. It is natural to try to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator, the latter constituting widely studied field of modern quantum physics. In this paper, it is proved that the deformed oscillators which realize quasi-boson operators in a consistent way really exist. The conditions for such realization are derived, and the uniqueness of the family of deformations under consideration is shown.

arXiv:1107.4297•Jul 21, 2011•2 citations

Composite Fermions as Deformed Oscillators: Wavefunctions and Entanglement

A. M. Gavrilik, Yu. A. Mishchenko

Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, п-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of composite Fermi-type particles, and explore them in two major cases: (i) “boson + fermion” composite fermions (or cofermions, or CFs); (ii) “deformed boson + fermion” CFs. As we show, cofermions in both cases admit only the realization by ordinary fermions. Case (i) is solved explicitly, and admissible wavefunctions are found along with entanglement measures. Case (ii) is treated within few modes both for CFs and constituents. The entanglement entropy and purity of CFs are obtained via the relevant parameters and illustrated graphically.

arXiv:1912.11021•Ukrainian Journal of Physics•Dec 9, 2019•3 citations

Entanglement entropy of composite fermions realized by (deformed) fermions vs. that of composite bosons

Alexandre M. Gavrilik, Yu. A. Mishchenko

In our two preceding papers we studied bipartite composite boson (or quasiboson) systems through their realization in terms of deformed oscillators. Therein, the entanglement characteristics such as the entanglement entropy and purity were found and expressed, for both one-quasiboson and more complex states, through the parameter of deformation. In this work we initiate an analogous study of composite fermions for two major cases: (i) "boson + fermion" composites; (ii) "deformed-boson + fermion" composites. Both the entanglement entropy and purity of composite fermions are dealt with, their dependence on the relevant parameters established, and for some particular two- or three-mode cases depicted graphically. In a few special cases the entanglement entropy turns out to be constant $S_0=\ln 2$ (or $\ln 3$) or $S_0=0$, while in the rest of the cases which we considered it varies between zero and $\ln 2$ (or $\ln 3$).

arXiv:1504.00529•Apr 2, 2015•2 citations

Two oscillators quantum groups and associated deformed coherent states

Nibaldo Alvarez-Moraga

Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates are computed giving a new class of deformed coherent states.

arXiv:math-ph/0503050•Mar 21, 2005•0 citations

A q-analogue of the supersymmetric oscillator and its q-supercoherent states

Gong Ren-shan

A q-analogue of the supersymmetric oscillator is constructed out of q-boson and ordinary fermion creation and annihilation operators. q-supercoherent states are explicitly obtained for the q-deformed supersymmetric oscillator. They are shown to be eigenstates of both q-boson and fermion annihilation operators and to satisfy a completeness relation. The representation of the q-deformed superalgebra in super Bargmann-Fock space is also discussed by means of the q-supercoherent states.

Journal of Physics A Mathematical and General•Oct 7, 1994•1 citation

q-deformed classical Lie algebras and their anyonic realization

M Frau, M A R-Monteiro, S Sciuto

All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.

arXiv:hep-th/9304108•Journal of Physics A Mathematical and General•Feb 7, 1994•9 citations

Composite bosons, here called quasibosons (e.g. mesons, excitons, etc), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the 'fermion+fermion' composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q -fermions ( q -deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is a quadratic polynomial in the number operator. In the case of two q -fermions, q ≠ 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.

The concept of quasi-bosons or composite bosons (like mesons, excitons etc.) has a wide range of potential physical applications. Even composed of two pure fermions, the quasi-boson creation and annihilation operators satisfy non-standard commutation relations. It is natural to try to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator, the latter constituting widely studied field of modern quantum physics. In this paper, it is proved that the deformed oscillators which realize quasi-boson operators in a consistent way really exist. The conditions for such realization are derived, and the uniqueness of the family of deformations under consideration is shown.

Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, п-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of composite Fermi-type particles, and explore them in two major cases: (i) “boson + fermion” composite fermions (or cofermions, or CFs); (ii) “deformed boson + fermion” CFs. As we show, cofermions in both cases admit only the realization by ordinary fermions. Case (i) is solved explicitly, and admissible wavefunctions are found along with entanglement measures. Case (ii) is treated within few modes both for CFs and constituents. The entanglement entropy and purity of CFs are obtained via the relevant parameters and illustrated graphically.

In our two preceding papers we studied bipartite composite boson (or quasiboson) systems through their realization in terms of deformed oscillators. Therein, the entanglement characteristics such as the entanglement entropy and purity were found and expressed, for both one-quasiboson and more complex states, through the parameter of deformation. In this work we initiate an analogous study of composite fermions for two major cases: (i) "boson + fermion" composites; (ii) "deformed-boson + fermion" composites. Both the entanglement entropy and purity of composite fermions are dealt with, their dependence on the relevant parameters established, and for some particular two- or three-mode cases depicted graphically. In a few special cases the entanglement entropy turns out to be constant

S_0=\ln 2

(or

\ln 3

) or

S_0=0

, while in the rest of the cases which we considered it varies between zero and

\ln 2

(or

\ln 3

).

Starting from a faithful five-dimensional matrix representation of the group of two independent oscillators and applying the R-matrix method we generate some classes of deformed fermionic-bosonic quantum Hopf algebras. The corresponding Lie deformed superalgebras of type I--II, obtained by duality, are computed and a realization of generators of these deformed superalgebras are given in terms of the usual fermionic and bosonic creation and annihilation operators associated to the supersymmetric harmonic oscillator. Then, a generalized deformed annihilator is construted and their eigenstates are computed giving a new class of deformed coherent states.

All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.