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Physical Review C

Dyson boson mapping of effective bi-fermion Hamiltonians

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

Published Mar 13, 2006• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physrevc.73.034306

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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, aˊla\'a\mathit{la}aˊla 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.
Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [12n{1}^{2n}12n]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.
The interaction between bi-fermionic and bosonic degrees of freedom is described by using an effective Hamiltonian inspired in QCD. The Dyson boson mapping technique is utilized in conjunction with a coherent state basis to construct energy landscapes in a multiparametric space. The appearance of fermionic and bosonic condensates is studied by minimizing the potential energy surface. The relationship between the domains of fermion and boson condensates and the order parameters of the model is discussed.
The spurious state problem is generalized Dyson boson mapping is examined. For cases where the fermion space is restricted to at most four-quasiparticle excitations, we show that Dyson boson images of normal-ordered quasiparticle interactions have identically vanishing matrix elements in the unphysical boson space. This ``decoupling'' feature explains the finding of previous authors that, in systems with at most four quasiparticles, every spurious state appears at its unperturbed energy. However, we show that the decoupling property does not generally hold in problems where there are excitations of more than four quasiparticles. Claims about methods for the spurious state problem, even though verified in four-quasiparticle calculations, may therefore be of limited validity.