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Finite boson mappings of fermion systems

Calvin W. Johnson, Joseph N. Ginocchio

Published Jul 1, 1994• 0 citations
OpenAlex

Abstract

We discuss a general mapping of fermion pairs to bosons that preserves Hermitian conjugation, with an eye towards producing finite and usable boson Hamiltonians that approximate well the low-energy dynamics of a fermion Hamiltonian.

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•Jul 1, 1994•0 citations
After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians, including effective operator theory to account for excluded states. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the Interacting Boson Model, but is more rigorous, general, and systematic.
Starting from a general, microscopic fermion-to-boson mapping that preserves Hermitian conjugation, we discuss truncations of the boson Fock space basis. We give conditions under which the exact boson images of finite fermion operators are also finite [e.g., a (1+2)-body fermion Hamiltonian is mapped to a (1+2)-body boson Hamiltonian] in the truncated basis. For the most general case, where the image is not necessarily exactly finite, we discuss how to make practical and controlled approximations.
Starting from a general, microscopic fermion-pair-to-boson mapping of a complete fermion space that preserves Hermitian conjugation, we show that the resulting infinite and non-convergent boson Hamilitonian can be factored into a finite (e.g., a 1 + 2-body fermion Hamiltonian is mapped to a 1 + 2-body boson Hamiltonian) image Hamilitonian times the norm operator, and it is the norm operator that is infinite and non-convergent. We then truncate to a collective boson space and we give conditions under which the exact boson images of finite fermion operators are also finite in the truncated basis.