Abstract
We are interested in the evaluation of the one- and two-body density matrices for extended boson matter. A trial function Ψ = Π i<j A⨍(r ij) of Bijl-Dingle-Jastrow form provides a reasonable first approximation to the ground state wave function of A bosons. The corresponding one-body density matrix has a structure n( r) = ρn c exp [− Q( r)] in the thermodynamic limit and shows off-diagonal long range order, n( r)→- ρn c > 0 as r→ ∞. The condensate fraction is given by n c = exp Q(0). A similar structure is found for the two-body density matrix, n( r 1, r 2, r' 1, r' 2) = χ( r) χ( r') exp[− R (r 1, r 2, r' 1,r' 2)] where r ≡ ¦r 1 − r 2¦, r' ≡ ¦r' 1 − r' 2¦ . The behavior over macroscopic distances is characterized by off-diagonal long range order of pairing type, n( r 1, r 2, r' 1, r' 2) → χ( r) χ( r') as ¦ r 1 − r' 1¦ → ∞ while r, r' < ∞ , in close analogy to the pairing phenomenon in Fermi superfluids. The pairing function χ( r) is related to the one-body density matrix by χ(r) = ⨍(r)n(r). The basic quantities Q( r) and Q(0) are analysed (i) in terms of a standard factor cluster expansion, (ii) in terms of a highly summed cluster expansion in the compact portions of the successive distribution functions g( r 12), g 3( r 1, r 2, r 3), …, (iii) within the hypernetted chain (HNC) framework. The HNC equations for Q( r) are solved analytically for uniform Bose fluids described by ¦g(r) − 1¦ ⪅ K ⪡ 1. We find in the uniform limit Q( r) = −(2 π) −3(4 ρ) −1 ∫ [ S( k) − 1] 2 S −1( k) exp (i k · r)d k where S( k) is the liquid structure function. The HNC formalism for quantities Q( r) and Q(0) is employed in a numerical study of the long-range order in (i) the charged boson system, (ii) a model of neutron matter, (iii) liquid 4He.
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