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Fortschritte der Physik/Progress of Physics

Density and Correlations in Large One and Three Dimensional Systems of Fermions

Naeem Ullah

Published Jan 1, 1989• 2 citations
Publisher

Abstract

The density and correlations are studied for a system of fermions moving in an average harmonic oscillator potential in one and three dimensions when the number of fermions are large. The technique is based on the use of Mehler's formula for the product of two Hermite functions and con-. tour integrations. Wigner transform is then calculated using this technique. It is shown that it becomes a step function when number of fermions become large. By integrating out momenta, an expression for the density of fermions is derived. Two-particle correlation function is studied in detail. Dyson's type of correlations for a system of nucleons is studied. The connection between one dimensional system of fermions and eigenvalues of a random Hermitean Hamiltonian is shown. The Coulomb energy difference of isobars is studied using nucleon density and correlations.

DOI
10.1002/prop.2190371103

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The density and correlations are studied for a system of fermions moving in an average harmonic oscillator potential in one and three dimensions when the number of fermions are large. The technique is based on the use of Mehler's formula for the product of two Hermite functions and con-. tour integrations. Wigner transform is then calculated using this technique. It is shown that it becomes a step function when number of fermions become large. By integrating out momenta, an expression for the density of fermions is derived. Two-particle correlation function is studied in detail. Dyson's type of correlations for a system of nucleons is studied. The connection between one dimensional system of fermions and eigenvalues of a random Hermitean Hamiltonian is shown. The Coulomb energy difference of isobars is studied using nucleon density and correlations.
We derive explicit results from an exact expression for fermion loops with an arbitrary number of density vertices in two dimensions at zero temperature, which has been obtained recently by Feldman et al. [in Singularities, edited by G. M. Greuel (Birkhaeuser, Basel, 1998)]. The 3-loop is an elementary function of the three external momenta and frequencies, and the N-loop can be expressed as a linear combination of 3-loops with coefficients that are rational functions of momenta and frequencies. We show that the divergencies of single loops for low energy and small momenta cancel each other when loops with permutated external variables are summed. The symmetrized N-loop, i.e., the connected N-point density-correlation function of the Fermi gas, does not diverge for low energies and small momenta. In the dynamical limit, where momenta scale to zero at fixed finite energy variables, the symmetrized N-loop vanishes as the $(2N\ensuremath{-}2)\mathrm{th}$ power of the scale parameter.
One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.
The properties of a certain class of unsymmetrized one-dimensional correlated Gaussian wave functions---those which are ground-state eigenfunctions of some coupled harmonic-oscillator Hamiltonian---are investigated in detail. It is shown that a properly symmetrized wave function constructed from these may be used to calculate the expectation value E0{E}_{0}E0​ of the Hamiltonian appropriate to a system of interacting one-dimensional atoms and that this energy is, to a high degree of accuracy, equal to the value obtained when the unsymmetrized wave function is used. A method is given by which correction terms to E0{E}_{0}E0​ may be obtained. In addition, it is found that even though the number of particles be very large, the necessary multivariable integrals may be performed quite simply.
We present explicit expressions for the correlation functions of interacting fermions in one dimension which are valid for arbitrary system sizes and temperatures. The result applies to a number of very different strongly correlated systems, including mesoscopic quantum wires, quantum Hall edges, spin chains and quasi-one-dimensional metals. It is, for example, possible to calculate Coulomb blockade oscillations from our expression and determine their dependence on interaction strength and temperature. Numerical simulations show excellent agreement with the analytical results.
Abstract This chapter introduces density operators and matrices for many-body systems, and explains their reduction to one- and two-body densities. Starting from the Slater determinant, the latter are explicitly calculated for independent fermions. For such systems, the classical limit is discussed and formulas for partition function, free energy, and entropy are derived by introducing and employing the concept of Wigner transformations. The propagation of wave packets is described in one dimension, a case for which correspondence rules are derived for various combinations of quantum operators. Finally, formulas are given for the equilibrium distribution of the quantal oscillator and the associated Wigner function.