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Physical review. B, Condensed matter

Pairing function for Bose fluids

MANFRED L. RISTIG

Published Aug 1, 1978• 10 citations
Publisher

Abstract

An elaborate cluster analysis of the pair occupancy ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ measures the simultaneous occupation of single-particle orbitals $\stackrel{^}{q}$, $\stackrel{^}{p}$ and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion ${\ensuremath{\chi}}_{q}^{2}$ for pairs of particles with equal and opposite momenta, $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\ensuremath{\chi}(r)$ which is defined as the Fourier inverse of quantity ${\ensuremath{\chi}}_{q}$ are displayed. It is further demonstrated that a close kinship exists between the function $\ensuremath{\chi}(r)$ and the one-particle density matrix $n(r)$. It is also shown that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlation factor implies a singular behavior $\ensuremath{\sim}{q}^{\ensuremath{-}1}$ of the function ${\ensuremath{\chi}}_{q}$ for small momenta $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$. The pairing function $\ensuremath{\chi}(r)$ is calculated numerically for the ground state of liquid $^{4}\mathrm{He}$ at equilibrium density adopting the familiar short-ranged correlation factor $f(r)$ employed by Schiff and Verlet and others.

DOI
10.1103/physrevb.18.1207

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Physical review. B, Condensed matter•Jul 1, 1982•18 citations
An elaborate cluster analysis of the pair occupancy {P}_{\stackrel{^}{q}\stackrel{^}{p}} is performed for a quantum fluid described by a Jastrow wave function. Quantity {P}_{\stackrel{^}{q}\stackrel{^}{p}} measures the simultaneous occupation of single-particle orbitals \stackrel{^}{q}, \stackrel{^}{p} and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion \ensuremathχq2{\ensuremath{\chi}}_{q}^{2}\ensuremathχq2​ for pairs of particles with equal and opposite momenta, \ensuremathℏq\ensuremath→+\ensuremathℏp\ensuremath→=0\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0\ensuremathℏq\ensuremath→​+\ensuremathℏp\ensuremath→​=0, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function \ensuremathχ(r)\ensuremath{\chi}(r)\ensuremathχ(r) which is defined as the Fourier inverse of quantity \ensuremathχq{\ensuremath{\chi}}_{q}\ensuremathχq​ are displayed. It is further demonstrated that a close kinship exists between the function \ensuremathχ(r)\ensuremath{\chi}(r)\ensuremathχ(r) and the one-particle density matrix n(r)n(r)n(r). It is also shown that the customary r\ensuremath−2{r}^{\ensuremath{-}2}r\ensuremath−2 long-range behavior of the two-body correlation factor implies a singular behavior \ensuremath∼q\ensuremath−1\ensuremath{\sim}{q}^{\ensuremath{-}1}\ensuremath∼q\ensuremath−1 of the function \ensuremathχq{\ensuremath{\chi}}_{q}\ensuremathχq​ for small momenta \ensuremathℏq\ensuremath→\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremathℏq\ensuremath→​. The pairing function \ensuremathχ(r)\ensuremath{\chi}(r)\ensuremathχ(r) is calculated numerically for the ground state of liquid 4He^{4}\mathrm{He}4He at equilibrium density adopting the familiar short-ranged correlation factor f(r)f(r)f(r) employed by Schiff and Verlet and others.
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.
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.
The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He^{4}\mathrm{He}4He. It is shown that Bose-Einstein condensation implies that the NNN-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r\mathbf{r}r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r\mathbf{r}r over a region of space proportional to VVV, where VVV is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.
Pair wave functions (PWF) which are eigenfunctions of the reduced density 2-matrix are considered for a homogeneous Bose liquid. With the Bogoliubov principle of the correlation weakening it is proved that the distribution of the "dissociated" pair states over momenta is exactly the product of the single-particle distribution functions. Thus, the "dissociated" pair states are naturally classified as condensate-condensate, condensate-supracondensate and supracondensate-supracondensate ones provided the Bose-Einstein condensate exists. The condensate-condensate as well as condensate-supracondensate PWF are expressed in terms of the averages of products of the creation and destruction Bose operators. This leads to the simple interpretation of the anomalous averages as the "scattering parts" of the condensate-condensate and condensate-supracondensate PWF. It is shown that in contrast to the Fermi liquid, the appearance of the anomalous averages for the Bose liquid does not necessarily mean that there exist bound states of pairs of particles. The PWF in the Hartree-Fock-Bogoliubov (HFB) approach are found. Given the density of the condensate is not zero, there are no bound pair states in the HFB scheme. The expansion of the pair correlation function in the set of PWF is very useful in order to take into account both short-range and long-range spatial correlations. Applications (possible and already realized) of the formalism developed are discussed.
The problem of how to extract information on the structure of the exact ground-state wave function of a Bose fluid from the results of Green's-function Monte Carlo computations is considered. Use of the criterion of maximum overlap of the exact ground state with an extended Jastrow function leads to certain equalities for correlation functions. By using methods borrowed from theories of classical fluids various schemes are proposed that permit the determination of the "best" Jastrow function. The case of Fermi statistics is also considered.