Exact, finite, and Hermitian fermion-pair-to-boson mapping

We discuss a general, exact (in that matrix elements are preserved) mapping of fermion pairs to bosons, and find a simple factorization of the boson representation of fermion operators. This leads to boson Hamiltonians that are Hermitian and finite, with no more than two-body operators if the fermion Hamiltonian has at most two-body operators, and one-body boson transition operators if the fermion transition operator is one-body. 03.65.Ca, 21.60.-n Typeset using REVTEX 1 Pairwise correlations are often important in describing the physics of many-fermion systems. The classic paradigm is the BCS theory of superconductivity [1], where the wavefunction is dominated by Cooper pairs which have electrons coupled up to zero linear momentum and spin; these boson-like pairs condense into a coherent wavefunction. Another example is the phenomenological Interacting Boson Model (IBM) for nuclei [2], where many states and transition amplitudes are successfully described using only sand d(angular momentum J = 0, 2) bosons, which represent coherent nucleon pairs. In both cases the large number of fermion degrees of freedom are well modelled by only a few boson degrees of freedom. On the other hand, however, despite some forays by Otsuka et al. [3], a rigorous microscopic basis for such phenomenological models is lacking. The basic problem is to represent the underlying fermion dynamics and statistics with a boson image amenable to approximation and numerical calculation. Considerable effort has gone into mapping fermion pairs into bosons [4,5]. However these mappings typically suffer from a variety of defects. Most, such as the Belyaev-Zelevinskii [6] and Marumori [7] mappings give rise to boson Hamiltonians with infinite expansions, that is, N -body terms with N → ∞. Convergence is slow even when “collective” fermion pairs are used. Finite but non-Hermitian boson Hamiltonians have also been derived [8,9]. In this Letter, using an alternate approach, we show that the infinite expansion boson Hamiltonian obtained from the exact mapping of fermion matrix elements to boson matrix elements can be factorized into a finite, Hermitian boson Hamiltonian times a norm operator, and it is the norm operator which has an infinite boson expansion. Consider a fermion space with 2Ω single-particle states, and a fermion Hamiltonian Ĥ. The general problem is to solve the fermion eigenvalue equation Ĥ |Ψp〉 = Ep |Ψp〉 , (1) find transition amplitudes between eigenstates, and so on. To do this we require a manybody basis. Often the basis set for many-fermion wavefunctions are Slater determinants, antisymmeterized products of single-fermion wavefunctions which we can write as products 2 of the Fock creation operators a†j , j = 1, · · ·, 2Ω on the vacuum a†i1 · · · a † in |0〉 for n fermions. These states span the antisymmetric irreducible representation of the unitary group in 2Ω dimensions, SU(2Ω). But for an even number of fermions one can just as well construct states from N = n/2 fermion pair creation operators, |ψβ〉 = N ∏ m=1 †βm |0〉 , (2) with †β ≡ 1 √ 2 ∑