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Physical review. B, Condensed matter

Statistical mechanics of strongly correlated Bose quantum fluids

B. E. Clements, E. Krotscheck, James A. Smith, C. E. Campbell

Published Mar 1, 1993• 26 citations
Publisher

Abstract

The finite-temperature properties of Bose quantum fluids are studied by applying a variational density-matrix approach. Our ansatz for the density matrix is similar to that introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. 30, 3728 (1984)] but has been generalized to include three-body correlations. An expression for the finite-temperature elementary-excitation energy, \ensuremath{\epsilon}(k), is derived from an ansatz for the variational entropy. At zero temperature \ensuremath{\epsilon}(k) reduces to the same expression obtained by Brillouin-Wigner perturbation theory. The hypernetted-chain (HNC) equations provide the relations between the n-particle distribution functions and the correlation functions. A substantial set of elementary diagrams have been included to minimize the loss of precision associated with the HNC approximation. We report values for the static structure function, radial distribution function, excitation energies, particle-hole interaction, Helmholtz free energy, entropy, and isothermal sound velocity. We present a detailed study of the spectra, three-body correlations, and elementary diagram dependence of the thermodynamic spinodal curve. For relevant densities, we find that three-body correlations tend to increase the liquid-gas critical temperature ${\mathit{T}}_{\mathit{c}}$ while adding elementary diagrams tends to reduce ${\mathit{T}}_{\mathit{c}}$. Including multiphonon terms in the spectrum produces several features in the variational entropy that were not observed at the Feynman level of excitations. Entropy isotherms show a rather abrupt transition from a low-density regime, where multiphonon contributions are insignificant, to a high-density regime where the isotherms are strongly influenced by multiphonon scattering processes.

DOI
10.1103/physrevb.47.5239

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The finite-temperature properties of Bose quantum fluids are studied by applying a variational density-matrix approach. Our ansatz for the density matrix is similar to that introduced by Campbell, K"urten, Ristig, and Senger [Phys. Rev. 30, 3728 (1984)] but has been generalized to include three-body correlations. An expression for the finite-temperature elementary-excitation energy, \ensuremath{\epsilon}(k), is derived from an ansatz for the variational entropy. At zero temperature \ensuremath{\epsilon}(k) reduces to the same expression obtained by Brillouin-Wigner perturbation theory. The hypernetted-chain (HNC) equations provide the relations between the n-particle distribution functions and the correlation functions. A substantial set of elementary diagrams have been included to minimize the loss of precision associated with the HNC approximation. We report values for the static structure function, radial distribution function, excitation energies, particle-hole interaction, Helmholtz free energy, entropy, and isothermal sound velocity. We present a detailed study of the spectra, three-body correlations, and elementary diagram dependence of the thermodynamic spinodal curve. For relevant densities, we find that three-body correlations tend to increase the liquid-gas critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ while adding elementary diagrams tends to reduce Tc{\mathit{T}}_{\mathit{c}}Tc​. Including multiphonon terms in the spectrum produces several features in the variational entropy that were not observed at the Feynman level of excitations. Entropy isotherms show a rather abrupt transition from a low-density regime, where multiphonon contributions are insignificant, to a high-density regime where the isotherms are strongly influenced by multiphonon scattering processes.
The variational density-matrix approach introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."
We study the temperature dependence of various spatial correlation functions of normal fluid and liquid 4He^{4}\text{H}\text{e}4He along an isochoric line of particle number density $\ensuremath{\varrho}=0.02185\text{ }{\text{\AA{}}}^{\ensuremath{-}3}$. The formal and numerical investigation employs correlated density-matrix theory as currently developed. The formalism permits a clear analysis of particle exchange (statistical) effects and so-called direct-direct quantum effects that appear only in a many-body system of interacting particles. At supercritical temperatures $T\ensuremath{\ge}12\text{ }\text{K}$ the helium fluid can be characterized as a perfect quantum Boltzmann system. In such a fluid, quantum statistical correlations are absent and the particle exchange follows classical Boltzmann statistics despite direct-direct quantum effects being present. At temperatures below 12 K, statistical quantum correlations begin to appear and are largest at the Bose-Einstein transition temperature TBE=2.17 K{T}_{\text{BE}}=2.17\text{ }\text{K}TBE​=2.17 K. However, these correlations are considerably smaller than the statistical exchange correlations existing in a free Bose gas of bosons with helium mass. We present analytical and numerical results on the radial distribution function, the exchange correlation function, the phase-phase correlation function, the one-body reduced density matrix, and gross thermal quantities such as specific heat and total exchange energy. The role of particle exchange in determining their quantum properties is analyzed and discussed in detail.
By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.
Within the framework of the variational theory for strongly interacting Bose liquids, we study the optimal determination of three-body correlation factors. The Euler-Lagrange equation for the three-body correlations suggests an iterative procedure in which the three-body correlation factor is expressed as a series of diagrams containing only the pair distribution function. The long-wavelength behavior of the three-body structure function agrees formally with the prediction of quantum hydrodynamics. The simplest approximation for the optimized three-body correlation function is identical to the prediction of second-order correlated-basis-function perturbation theory. We show that this simplest approximation sums essentially the same sets of diagrams that are included in non-optimized hypernetted-chain calculations with three-body correlations. Along with the three-body correlations we include those elementary diagrams of fourth and fifth order which have a comparable topological structure. The predictions for the ground-state energy and the pair distribution function are substantially improved compared with hypernetted-chain calculations restricted to two-body correlations.
The energy and structure of dilute gases of hard spheres in three dimensions is discussed, together with some aspects of the corresponding 2D systems. A variational approach in the framework of the Hypernetted Chain Equations (HNC) is used starting from a Jastrow wavefunction that is optimized to produce the best two–body correlation factor with the appropriate long range. Relevant quantities describing static properties of the system are studied as a function of the gas parameter x = ρa d where ρ, a and d are the density, s–wave scattering length of the potential and dimensionality of the space, respectively. The occurrence of a maximum in the radial distribution function and in the momentum distribution is a natural effect of the correlations when x increases. Some aspects of the asymptotic behavior of the functions characterizing the structure of the systems are also investigated. PACS numbers: 03.75.Hh, 05.30.Jp, 67.40.Db. 1.