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arXiv:2106.07839 · Lecture notes series, Institute For Mathematical Sciences

DENSITY-POTENTIAL FUNCTIONAL THEORY FOR FERMIONS IN ONE DIMENSION

Martin-Isbjörn Trappe, Jun Hao Hue, Berthold‐Georg Englert

Published Feb 26, 2023• 2 citations
arXivHTMLPDF

Abstract

We showcase the advantages of orbital-free density-potential functional theory (DPFT), a more flexible variant of Hohenberg-Kohn density functional theory. DPFT resolves the usual trouble with the gradient-expanded kinetic energy functional by facilitating systematic semiclassical approximations in terms of an effective potential energy that incorporates all interactions. With the aid of two systematic approximation schemes we demonstrate that DPFT is not only scalable, universally applicable in both position and momentum space, and allows kinetic and interaction energy to be approximated consistently, but can also compete with highly accurate, yet restricted, methods. As two- and three-dimensional geometries are extensively covered elsewhere, our focus here is on one-dimensional settings, with semiclassical observables systematically derived from both the Wigner function formalism and a split-operator approach. The high quality of our results for Fermi gases in Morse potentials invites the use of DPFT for describing more exotic systems, such as trapped large-spin fermion mixtures with contact or dipole-dipole interactions.

DOI
10.1142/9789811272158_0008

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We showcase the advantages of orbital-free density-potential functional theory (DPFT), a more flexible variant of Hohenberg-Kohn density functional theory. DPFT resolves the usual trouble with the gradient-expanded kinetic energy functional by facilitating systematic semiclassical approximations in terms of an effective potential energy that incorporates all interactions. With the aid of two systematic approximation schemes we demonstrate that DPFT is not only scalable, universally applicable in both position and momentum space, and allows kinetic and interaction energy to be approximated consistently, but can also compete with highly accurate, yet restricted, methods. As two- and three-dimensional geometries are extensively covered elsewhere, our focus here is on one-dimensional settings, with semiclassical observables systematically derived from both the Wigner function formalism and a split-operator approach. The high quality of our results for Fermi gases in Morse potentials invites the use of DPFT for describing more exotic systems, such as trapped large-spin fermion mixtures with contact or dipole-dipole interactions.
Density-potential-functional theory (DPFT) is an alternative formulation of orbital-free density functional theory that may be suitable for modeling the electronic structure of large systems. To date, DPFT has been applied mainly to quantum gases in one- and two-dimensional settings. In this work, we study the performance of DPFT when applied to real-life systems: atoms, dimers, and nanoparticles. We build on systematic Suzuki-Trotter factorizations of the quantum-mechanical propagator and on the Wigner function formalism, respectively, to derive nonlocal as well as semilocal functional approximations in complete analogy to their well-established lower-dimensional versions, without resorting to system-specific approximations or ad hoc measures of any kind. The cost for computing the associated semiclassical ground-state single-particle density scales (quasi)linearly with particle number. We illustrate that the developed density formulas become relatively more accurate for larger particle numbers, can be improved systematically, are quite universally applicable, and, hence, may offer alternatives to existing orbital-free methods for mesoscopic quantum systems.
We present an alternative to the Kohn-Sham formulation of density-functional theory for the ground-state properties of strongly interacting electronic systems. The idea is to start from the limit of zero kinetic energy and systematically expand the universal energy functional of the density in powers of a "coupling constant" that controls the magnitude of the kinetic energy. The problem of minimizing the energy is reduced to the solution of a strictly correlated electron problem in the presence of an effective potential, which in our theory plays the same role as the Kohn-Sham potential plays in the traditional formulation. We discuss several schemes for approximating the energy functional, and report preliminary results for low-density quantum dots.
The semiclassical $\ensuremath{\hbar}$ expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of B. Grammaticos and A. Voros [Ann. Phys. (N.Y.) 123, 359 (1979)], originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virtue of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\ensuremath{\hbar}$ expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a finite, second-order gradient correction to the Hartree-Fock energy, which takes the form $\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}$, with $\ensuremath{\varepsilon}$ being small ($|\ensuremath{\varepsilon}|\ensuremath{\ll}1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.
Two of the most popular quantum-mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a one-dimensional (1D) double-well potential, a model of atoms in a quasi-1D optical lattice, or electrons of a hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density-functional theory (DFT) functional and two-site Hubbard models. A 1D adiabatic local-density approximation kernel is constructed for use in time-dependent density-functional theory (TDDFT) and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably to the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with nonadiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.
In a recent paper [B. P. van Zyl et al., Phys. Rev. A 89, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density functional theory of an inhomogeneous, two-dimensional (2D) Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density functional theory and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, without any gradient corrections, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor pointwise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for both the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.
We develop an functional framework to compute one-body quasiprobabilities for both fermionic and bosonic systems. Since the key variable is a quasidensity, this theory circumvents the problems of finding the Pauli potential or approximating the kinetic energy that are known to limit the applicability of standard orbital-free density functional theory. We present a set of strategies to (a) compute the one-body Wigner quasiprobability in an orbital-free manner from the knowledge of the universal functional and (b) obtain those functionals from the functionals of the one-body reduced density matrix (1-RDM). We find that the universal functional of optical lattices results from a translation, a contraction, and a rotation of the corresponding functional of the 1-RDM, revealing the strong connection between these two functional theories. Furthermore, we relate the key concepts of and . Published by the American Physical Society 2024