Correlated Density Matrix Research Articles

Correlated Density Matrix Theory of Boson Superfluids

A variational approach to unified microscopic description of normal and superfluid phases of a strongly interacting Bose system is proposed. We begin the formulation of an optimal theory within this approach through the diagrammatic analysis and synthesis of key distribution functions that characterize the spatial structure and the degree of coherence present in the two phases. The approach centers on functional minimization of the free energy corresponding to a suitable trial form for the many-body density matrix W(R, R′) ∝ Φ(R) Q(R, R′) Φ(R′), with the wave function Φ and incoherence factor Q chosen to incorporate the essential dynamical and statistical correlations. In earlier work addressing the normal phase, Φ was taken as a Jastrow product of two-body dynamical correlation factors and Q was taken as a permanent of short-range two-body statistical bonds. A stratagem applied to the noninteracting Bose gas by Ziff, Uhlenbeck, and Kac is invoked to extend this ansatz to encompass both superfluid and normal phases, while introducing a variational parameter B that signals the presence of off-diagonal long-range order. The formal development proceeds from a generating functional Λ, defined by the logarithm of the normalization integral ∫ dR Φ2(R) Q(R, R). Construction of the Ursell-Mayer diagrammatic expansion of the generator Λ is followed by renormalization of the statistical bond and of the parameter B. For B ≡ 0, previous results for the normal phase are reproduced, whereas For B > 0, corresponding to the superfluid regime, a new class of anomalous contributions appears. Renormalized expansions for the pair distribution function g(r) and the cyclic distribution function Gcc(r) are extracted from Λ by functional differentiation. Standard diagrammatic techniques are adapted to obtain the appropriate hypernetted-chain equations for the evaluation of these spatial distribution functions. Corresponding results are presented for the internal energy. The quantity Gcc(r) is found to develop long-range order in the condensed phase and therefore, assumes an incisive diagnostic role in the elucidation of the Bose-Einstein transition of the interacting system. A tentative connection of the microscopic description with the phenomenological two-fluid model is established in terms of a sum rule on "normal" and "anomalous" density components. Further work within this correlated density matrix approach will address the one-body density matrix, the entropy, and the Euler-Lagrange equations that lead to an optimal theory.

The surface of liquid4He at nonzero temperatures

We outline a general approach to microscopic evaluation of the properties of strongly interacting, spatially inhomogeneous Bose systems at finite temperatures. A minimum principle for the Helmholtz free energy is used together with an appropriate trial density matrix to generalize the correlated variational wave function theory that has proven so successful in the treatment of the ground states and elementary excitations of quantum fluids at zero temperature. Euler-Lagrange equations are obtained that determine the optimal structure through the one-and two-body densities and the optimal density fluctuation operators and energies characterizing the elementary excitations. Some results of an application of this correlated density matrix theory to the4He liquid-vapor interface are presented, with particular focus on the characterization of resonant vapor modes.

Temperature dependence of collective excitations in liquid deuterium studied by neutron inelastic scattering

The inelastic response from liquid deuterium has been measured on a thermal three-axis neutron spectrometer at two different temperatures. The observed neutron scattering spectra show clear inelastic features which have been analysed using a damped harmonic oscillator (DHO) model. The momentum transfer dependences of the DHO frequencies and damping coefficients at each temperature are compared and related to the results of the correlated density matrix formalism.

Collective excitations in liquid deuterium in three thermodynamic states

The coherent inelastic response from liquid deuterium in three different thermodynamic states has been measured on a thermal three axis neutron spectrometer. The observed dispersive behaviour has been analyzed using a damped harmonic oscillator model. The momentum transfer dependencies of the renormalized frequencies and damping coefficients are compared and related to the results from correlated density matrix formalism.

Correlated density matrix theory of normal quantum fluids

A variational description of the normal phase of a strongly interacting N-boson fluid at finite temperature is formulated in terms of trial density matrices designed to incorporate the essential dynamical and statistical correlations. The approach followed is a natural extension, to the noncondensed phase, of the familiar Jastrow-Feenberg variational approximation of correlated-basis functions theory, which has provided a firm foundation for microscopic calculations of the properties of liquid 4He at zero temperature. The N-body density matrix elements of the homogeneous fluid are represented in the general form W(R, R′) = J −1 φ(R) Q(R, R′) φ(R′) involving a normalization integral J = ∫ dR φ 2(R) Q(R, R), where R denotes a point in the configuration space of the system. This form is specialized by assuming φ( R) to be a wave function of Jastrow type, i.e., a product of two-body dynamical correlation functions f(r) = exp[ u(r) 2 ] , and taking Q( R, R′) to be a permanent part of two-body statistical functions Γ( r). It is in the latter assumption, which tailors the treatment to the normal phase, that the present approach departs from earlier work within variational density matrix theory, which focuses instead on the condensed or superfluid phase. The internal energy, the spatial distribution functions, and the one-body density matrix of the system may all be constructed from variational derivatives of the normalization integral J (or of a generalized version of J ), carried out with respect to the dynamical or statistical correlation functions entering the theory. The required normalization integrals are susceptible to analysis in terms of generalized Ursell-Mayer cluster diagrams; systematic diagrammatic resummations based on the hypernetted-chain (HNC) classification scheme are then performed following standard prescriptions. Neglecting elementary diagrams, one arrives at explicit expressions for the indicated equilibrium properties as functionals of the trial correlations u( r) and Γ( r). Making use of the replica technique, it is possible to treat the entropy by the same procedures. With appropriate simplifications to facilitate the associated analytic continuation, one arrives at an entropy expression equivalent to that of a system of noninteracting bosons occupying single-particle momentum states k with an occupation number n cc ( k) determined by the solution of the HNC equations arising in the treatment of the internal energy. In the culminating step of the formal development, an explicit functional expression for the Helmholtz free energy is assembled from the results obtained for the internal energy and entropy, and the Gibbs-Delbrück-Molière minimum principle is invoked. Functional variation of the free energy with respect to the correlation function u( r) and a renormalized statistical correlation function Γ cc ( r) leads to coupled Euler-Lagrange equations which are the analogs of the paired-phonon equation for a Bose superfluid at zero temperature and of the Feynman eigenvalue equation determining the elementary excitations of the superfluid phase. The solutions provide the ingredients of a condition that signals the occurrence of a Bose-Einstein transition in the correlated system. Specific attention is given to the inclusion of phonon effects in the ansatz for the correlated density matrix, a necessary refinement for quantitative treatment of the gas-liquid spinodal line and critical point. An analogous correlated density matrix formalism is developed for the normal phase of a strongly interacting fluid of spinless fermions at finite temperature. In the limiting case of zero temperature, this approach reproduces the well-known Fermi hypernetted-chain theory of the Jastrow-correlated Fermi sea.

Equation for the direct determination of the density matrix

A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\odinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.