Correlated Density Matrix (CDM) theory permits formal analyses of microscopic properties of strongly correlated quantum fluids and liquids at nonzero temperatures. Equilibrium properties, thermodynamic potentials, correlation and structure functions can be studied formally as well as numerically within the CDM algorithm. Here we provide the essential building blocks for studying the radial distribution function and the single-particle momentum distribution of the ingredients of the quantum systems. We focus on the statistical properties of correlated fluids and introduce the concept of renormalized bosons and fermions. These entities carry the main statistical features of the correlated systems such as liquid 4 He through their specific dependence on temperature, particle number density, and wavenumber encapsulated in their effective masses. The formalism is developed for systems of bosons and of fermions. Numerical calculations for fluid 4 He in the normal phase demonstrate the power of the renormalization concept. The formalism is further extended to analyze the Bose-Einstein condensed phases and gives a microscopic understanding of Tisza's two-fluid model for the normal and superfluid density components.
Correlated Density Matrix (CDM) theory permits formal analyses of microscopic properties of strongly correlated quantum fluids and liquids at nonzero temperatures. Equilibrium properties, thermodynamic potentials, correlation and structure functions can be studied formally as well as numerically within the CDM algorithm. Here we provide the essential building blocks for studying the radial distribution function and the single-particle momentum distribution of the ingredients of the quantum systems. We focus on the statistical properties of correlated fluids and introduce the concept of renormalized bosons and fermions. These entities carry the main statistical features of the correlated systems such as liquid 4 He through their specific dependence on temperature, particle number density, and wavenumber encapsulated in their effective masses. The formalism is developed for systems of bosons and of fermions. Numerical calculations for fluid 4 He in the normal phase demonstrate the power of the renormalization concept. The formalism is further extended to analyze the Bose-Einstein condensed phases and gives a microscopic understanding of Tisza's two-fluid model for the normal and superfluid density components.
Correlated density matrix theory is generalized to investigate equilibrium properties of normal Fermi Liquids such as 3He and nuclear matter at nonzero temperatures. The results also generalize the Fermi-hypernetted-chain technique that is familiar from studies of the ground state of correlated fermions. By employing the concept of renormalized bosons and fermions the formal results are cast in a form that permits the direct evaluation of the statistical properties of the correlated liquid such as the entropy and the specific heat at constant volume among other quantities.
We study the temperature dependence of various spatial correlation functions of normal fluid and liquid 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=2.17K. 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.
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 while adding elementary diagrams tends to reduce 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.
A useful concept for finding numerically the dominant correlations of a given ground state in an interacting quantum lattice system in an unbiased way is the correlation density matrix (CDM). For two disjoint, separated clusters, it is defined to be the density matrix of their union minus the direct product of their individual density matrices and contains all the correlations between the two clusters. We show how to extract from the CDM a survey of the relative strengths of the system's correlations in different symmetry sectors and the nature of their decay with distance (power law or exponential), as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. To achieve this goal, we introduce a new method of analysing the CDM, termed the dominant operator basis (DOB) method, which identifies in an unbiased fashion a small set of operators for each cluster that serve as a basis for the dominant correlations of the system. We illustrate this method by analysing the CDM for a spinless extended Hubbard model that features a competition between charge density correlations and pairing correlations, and show that the DOB method successfully identifies their relative strengths and dominant correlators. To calculate the ground state of this model, we use the density matrix renormalization group, formulated in terms of a variational matrix product state (MPS) approach within which subsequent determination of the CDM is very straightforward. In an extended appendix, we give a detailed tutorial introduction to our variational MPS approach for ground state calculations for one-dimensional quantum chain models. We present in detail how MPSs overcome the problem of large Hilbert space dimensions in these models and describe all the techniques needed for handling them in practice.
This lecture describes certain properties of interacting Bose gases and superconductors which have recently been considered at Harvard. It is a brief resumé of work by the lecturer, P.C. Hohenberg, A. Fetter, R. Lange, and C. De Dominicis, which will be reported in mathematical detail elsewhere. It concerns attempts to derive from fundamental principles, several aspects of the macroscopic two-fluid model of London, Landau, and Tisza, the Landau phonon spectrum for bosons, and the structured condensate envisaged by Onsager and Feynman. The lecture describes new results which have been obtained concerning: (1) methods for calculating properties of large, but not necessarily infinite, condensed systems at finite temperatures; (2) methods for performing calculations in boson systems consistent with current conservation and sum rules; (3) results obtained by these methods for the phonon spectrum; (4) more accurate results for the less physical single-particle spectrum at finite temperatures; (5) results obtained for the macroscopically structured condensate in rotating systems which agree with the macroscopically inferred results of Feynman and Onsager; (6) results which treat depletion of the condensate as a result of interaction consistently and verify that nonetheless, the superfluid density, as operationally determined, is not correspondingly depleted; (7) methods in which the entropy plays a central role, for casting the macroscopic theory in a form which lays stress on the renormalized excitations of renormalized interactions; (8) the almost identical features of the mathematics for condensed Fermi and Bose systems; (9) a restatement, in this microscopic transcription, of the Feynman-Onsager argument for flux and vorticity quantization.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).