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Physical review. B, Solid state

Application of field-theoretic, collective-coordinate, and correlated-basis-function methods to many-boson systems

Fereydoon Family, Harvey Gould

Published Nov 1, 1975• 5 citations
Publisher

Abstract

The collective-coordinate and correlated-basis-function approaches are systematically applied to the determination of the elementary excitation spectrum $\ensuremath{\omega}(k)$ and the static structure function $S(k)$ for two models of a Bose gas which can be characterized by a single small expansion parameter $g$. Results are obtained to first order in $g$, the leading order in which multiexcitation processes are present, and are compared with those found by the microscopic-dielectric-function approach. It is found that a collective-coordinate calculation of $\ensuremath{\omega}(k)$ by Sunakawa, Yamasaki, and Kebukawa is incomplete to first order in $g$ and that the convolution approximation for the three-particle distribution function in the correlated-basis-function formalism leads to incorrect results for overlap matrix element and for $S(k)$. A form for the overlap matrix element is proposed which leads to correct first-order results. Various properties of $\ensuremath{\omega}(k)$, $S(k)$, and the dynamic structure function are discussed.

DOI
10.1103/physrevb.12.3729

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The collective-coordinate and correlated-basis-function approaches are systematically applied to the determination of the elementary excitation spectrum \ensuremathω(k)\ensuremath{\omega}(k)\ensuremathω(k) and the static structure function S(k)S(k)S(k) for two models of a Bose gas which can be characterized by a single small expansion parameter ggg. Results are obtained to first order in ggg, the leading order in which multiexcitation processes are present, and are compared with those found by the microscopic-dielectric-function approach. It is found that a collective-coordinate calculation of \ensuremathω(k)\ensuremath{\omega}(k)\ensuremathω(k) by Sunakawa, Yamasaki, and Kebukawa is incomplete to first order in ggg and that the convolution approximation for the three-particle distribution function in the correlated-basis-function formalism leads to incorrect results for overlap matrix element and for S(k)S(k)S(k). A form for the overlap matrix element is proposed which leads to correct first-order results. Various properties of \ensuremathω(k)\ensuremath{\omega}(k)\ensuremathω(k), S(k)S(k)S(k), and the dynamic structure function are discussed.
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The validity of Sunakawa, Yamasaki and Kebukawa's Hamiltonian and that of Bogoliubov and Zubarev's Hamiltonian are examined. Perturbational expansion of the ground state energy by these Hamiltonians disagrees with the exact solution of Lieb and Liniger for one-dimensional Bose system with repulsive delta-function interaction. This fact suggests that these Hamiltonians are not microscopic descriptions of the many-Boson system. Mathematical inconsistency in Bogoliubov and Zubarev's theory is also pointed out. Moreover analytic expression of high density expansion for the ground state energy density e0 is found out to be e0n-3=γ-(4/3π)γ3/2+(1/6-1/π2)γ2+O(γ5/2), γ≡c/n, for one-dimensional Bose system with delta function interaction (density n, strength 2c, ħ=2m=1) by the use of the correlated basis function method.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid He3{\mathrm{He}}^{3}He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$. FFF is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the f(rij)f({r}_{\mathrm{ij}})f(rij​) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​, Nmn{\mathfrak{N}}_{\mathrm{mn}}Nmn​ of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​ are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​) and (Nmn{\mathfrak{N}}_{\mathrm{mn}}Nmn​), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (Hmn{H}_{\mathrm{mn}}Hmn​) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (Hmn{H}_{\mathrm{mn}}Hmn​). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (Hmn{H}_{\mathrm{mn}}Hmn​). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any FFF in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}, QQQ-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid He3{\mathrm{He}}^{3}He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.
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