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Physica

Cluster developments for Jastrow wave functions II Introduction of irreducible cluster functions

Chr.D. Hartogh, H.A. Tolhoek

Published Jan 1, 1958• 16 citations
Publisher
DOI
10.1016/s0031-8914(58)80105-6

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CLUSTER DEVELOPMENT OF THE DENSITY MATRICES IN THE JASTROW METHOD FOR MANY- FERMION SYSTEMS

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A method is given for the cluster development of the reduced density matrices for Jastrow wave functions. The method makes extensive use of the properties of the Dirac densitv matrices corresponding to the uncorrelated niotion, in conjunction with the method of expanding ihe distribution functions for a classical imperfect gas given by Maver and Montroll. (auth)

Arkiv Fysik•Jan 1, 1960•0 citations
An elaborate cluster analysis of the single-particle occupation probability ${n}_{\stackrel{^}{q}}$ and associated one-body density matrix n(r)n(r)n(r) is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that ${n}_{\stackrel{^}{q}}$ may be written as n[N(q)+N1(q)]n[N(q)+{N}_{1}(q)]n[N(q)+N1​(q)], where nnn is a strength factor independent of wave number qqq and the quantities N(q)N(q)N(q) and N1(q){N}_{1}(q)N1​(q) may be expressed as series of irreducible cluster contributions. The strength factor nnn has the form n=eQn={e}^{Q}n=eQ, where QQQ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for QQQ in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of ${n}_{\stackrel{^}{q}}$, it is further demonstrated that n(r)n(r)n(r) may be cast in the form $\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)$, where $\ensuremath{\rho}$ is the particle density and the functions N1(r){N}_{1}(r)N1​(r), N2(r){N}_{2}(r)N2​(r), and Q(r)\mathcal{Q}(r)Q(r) are all given by irreducible cluster series. Massive partial summations are executed in the Q(r)\mathcal{Q}(r)Q(r) series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has Q(0)=Q\mathcal{Q}(0)=QQ(0)=Q. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid 3He^{3}\mathrm{He}3He and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid 4He^{4}\mathrm{He}4He to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}$ being just the strength factor nnn. It may also be shown (under certain reasonable assumptions) that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlations implies a singular behavior ${n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}$ of the Bose momentum distribution for small qqq.