Generalised Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalised Dyson boson-fermion mapping is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that represents a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that it will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this text are to review the theory of generalized Dyson boson-fermion mappings and to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. The Dyson mapping uncovers non-trivial properties of the system that aid the construction of time-independent as well as time-dependent perturbation expansions. The time-independent expansions agree with results that other authors have obtained through methods other than boson expansions. The time-dependent expansions might in future prove useful in understanding aspects of the dynamics of ultra-cold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.Generalised Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalised Dyson boson-fermion mapping is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that represents a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that it will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this text are to review the theory of generalized Dyson boson-fermion mappings and to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. The Dyson mapping uncovers non-trivial properties of the system that aid the construction of time-independent as well as time-dependent perturbation expansions. The time-independent expansions agree with results that other authors have obtained through methods other than boson expansions. The time-dependent expansions might in future prove useful in understanding aspects of the dynamics of ultra-cold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.

arXiv:nucl-th/0411059Nov 16, 20040 citations

Spurious states in boson mapping

Cheng-Li Wu, Wei-Xin Tang, D.H. Feng

Physics Letters BMay 1, 19854 citations

Dyson boson mapping of effective bi-fermion Hamiltonians

Osvaldo Civitarese, H. B. Geyer, M. Reboiro

Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, $\'a\mathit{la}$ Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, $\'a\mathit{la}$ Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.

Physical Review CMar 13, 20061 citation

\'a\mathit{la}

Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.