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The Journal of Chemical Physics

Relations between parameters of density functional theories through exactly solvable many-fermion models

A. R. P. Rau, Jürgen Hinze

Published Jun 8, 2001• 1 citation
Publisher

Abstract

By considering exactly solvable many-fermion models in different spatial dimensions and with different interaction potentials, the chemical potential and Lagrange multiplier for the normalization requirement are identified with the energy of the last particle, and their relation to the average energy per particle is established.
DOI
10.1063/1.1371499

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In the unitary regime, fermions interact strongly via two-body potentials that exhibit a zero range and a (negative) infinite scattering length. The energy density is proportional to the energy density of the free Fermi gas with a proportionality constant $\ensuremath{\xi}$. The author uses a simple density functional parametrized by an effective mass and the universal constant $\ensuremath{\xi}$, and employs Kohn-Sham density-functional theory to obtain the parameters from fit to one exactly solvable two-body problem. This yields $\ensuremath{\xi}=0.42$ and a rather large effective mass. The form of the density functional is checked by similar Kohn-Sham calculations for the exactly solvable Calogero model.
In the framework of exactly solvable quantum many-body systems we study models of interacting spin one-half Fermions in one dimension. The first part deals with systems of spin one-half Fermions which interact via repulsive contact interaction. A reformulation of the Bethe-Ansatz solvable many-body wave function is presented. This simplifies considerably the calculations for the highly imbalanced case, where very few particles of one species (minority Fermions) are present. For the other particle species (majority Fermions) the thermodynamic limit is taken. We assume the majority Fermions to be in the ground state such that their non-interacting momentum distribution is a Fermi-sea. Upon this we consider excitations where the particles of the minority species may occupy an arbitrary state within the Fermi-sea. In the case of only a single minority Fermion, the many-body wave function can be expressed as a determinant. This allows us to derive exact thermodynamic expressions for several expectation values as well as for the density-density correlation function. Moreover it is possible to find closed expressions for the single particle Green's function. All of the above mentioned quantities show a non-trivial dependence on the minority particle's momentum. In particular the Green's function in the Tonks-Girardeau regime of hardcore interaction is shown to undergo a transition from the one of impenetrable Bosons to that of free Fermions as the extra particle's momentum varies from the core to the edge of the Fermi-sea. This transition becomes manifest in an algebraic asymptotic decay of the Green's function. If two minority Fermions are present, the many-body wave function turns out to be more complicated. Nevertheless it is possible to derive exact expressions for the two and the three particle density-density correlation functions. Furthermore we calculate the system's total energy and based on that, identify terms which have a natural interpretation as effective interaction energy for the two minority Fermions in the presence of the Fermi-sea. The second part is devoted to the study of one-dimensional systems consisting of two fermionic particle species with different masses. We show that for specific kinds of interaction potentials and for certain relations between the masses and the coupling constants, the particle creation and annihilation operators of such a system can be constructed exactly.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.
Remarkably the one-dimensional (1D) many-fermion fluid with pairwise-attractive \ensuremath{\delta}-function interactions is exactly solvable in that one can determine the exact many-body ground-state energy and chemical potential for all values of the coupling strength and/or density. Bardeen-Cooper-Schrieffer (BCS) theory is tested in this model by numerically determining the BCS total ground-state energy and chemical potential as a function of the coupling strength and/or density, and comparing with the exact results. As is the case for 2D and 3D theories, two regimes are apparent: (a) a BCS-proper regime of weakly coupled, overlapping Cooper pairs and (b) a Bose-gas regime of strongly interacting fermions which pair to form an ideal Bose gas at low density. In the two extremes the BCS energy and chemical potential are identical to the exact values and are moderately close for intermediate coupling and/or density.
A recently proposed Boltzmann local equilibrium Wigner function for massive spin-1/2 particles is generalized to the case of Fermi-Dirac statistics. The resulting formula ensures the correct normalization of the mean polarization vector and reproduces the generalized thermodynamic relations with spin that were obtained in earlier studies. Moreover, we show that the macroscopic currents constructed from the Fermi-Dirac Wigner function can be obtained as derivatives of a suitably defined generating function with respect to the Lagrange multipliers (temperature, hydrodynamic flow, and chemical potentials). The identified generating function also indicates that the underlying framework can be classified as a divergence-type theory.