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Europhysics Letters (EPL)

Spurious Components in the Ideal-Boson States at Thermal-Field Boson Realization

Р.Б. Бегжанов, E. M. Galinsky

Published Aug 1, 1990• 1 citation
Publisher

Abstract

A Dyson boson realization for two-particle operators at finite temperature is derived using the thermal-field dynamic formalism. The nature of spurious state appearance in the ideal-boson states is investigated. It is further observed that the matrix element of the pairing interaction between the ideal-boson basis is identical to the corresponding fermion matrix element.

DOI
10.1209/0295-5075/12/7/008

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A Dyson boson realization for two-particle operators at finite temperature is derived using the thermal-field dynamic formalism. The nature of spurious state appearance in the ideal-boson states is investigated. It is further observed that the matrix element of the pairing interaction between the ideal-boson basis is identical to the corresponding fermion matrix element.
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.
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.
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.
We present an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T .This is achieved in the framework of the real time formalism of Thermofield Dynamics.The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature.In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermioncurrent two-point function in the imaginary-time formalism and real time formalisms.
An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.