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Journal of Scientific Computing

Projected Sobolev Gradient Flows for Computing Ground State of Ultracold Dipolar Fermi Gas Based on Density Functional Theory

Xuelin Zhang, Chunping Pang, Hanquan Wang

Published Oct 3, 2025• 0 citations
Publisher
DOI
10.1007/s10915-025-03076-9

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Abstract. In this paper, a gradient flow model is presented for conducting ground state calculations in the Wigner formalism of many-body systems in the framework of density functional theory. Theoretically, an energy functional in the Wigner formalism is proposed, based on which the minimization problem is designed and analyzed for the ground state, providing a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is built upon the energy functional to obtain the ground state Wigner function representing the entire many-body system. Numerically, a parallelizable algorithm is developed using the operator splitting method and the Fourier spectral collocation method, whose numerical complexity of a single iteration is [Formula: see text]. Numerical experiments demonstrate the anticipated accuracy, encompassing the one-dimensional system with up to [Formula: see text] particles and the three-dimensional system with defect, showcasing the potential of our approach to large-scale simulations and computations of systems with defect. To the best of our knowledge, this is the first deterministic method for calculating the ground state Wigner function of the three-dimensional many-body system.

arXiv:2409.10851•SIAM Journal on Scientific Computing•Sep 12, 2025•0 citations
Abstract The spin-polarized 2D dipolar Fermi gas trapped in a harmonic potential provides an opportunity to study the many-fermion systems with strong particle–particle correlation in a controlled fashion. In this paper, we investigate the role of the correlation part of the interaction energy on the ground-state properties and on the collective oscillation frequencies of a 2D dipolar Fermi gas by employing density functional theory (DFT) within local density approximation. We employ a purely density-based, orbital-free approach of DFT to study the ground-state properties. We employ an orbital-free approach, as it is suitable for handling a large number of fermions (10 3 –10 4 atoms/molecules), which are typically the number of atoms/molecules contained in the samples of dipolar Fermi gas created in the laboratories. We derive analytical expressions for the frequencies of the monopole and quadrupole modes of the collective oscillations by employing a method based on the sum-rule approach of linear response theory. We find that the exchange and correlation part of the fermion–fermion interaction contributes significantly to the total energy, and their contributions increase with increasing number of fermions and interaction strength. In particular, the correlation part lowers the total energy of the 2D dipolar fermionic system, thereby stabilizing the system. Similarly, the inclusion of the correlation effect lowers the frequencies of the monopole mode of the collective oscillations appreciably in comparison to the case when this effect is not taken into account. On the other hand, the frequency of the quadrupole mode of the collective oscillations is not appreciably altered by the correlation part of the interaction over a wide range of values of interaction strength and number of fermions.
Abstract. In this paper, a gradient flow model is presented for conducting ground state calculations in the Wigner formalism of many-body systems in the framework of density functional theory. Theoretically, an energy functional in the Wigner formalism is proposed, based on which the minimization problem is designed and analyzed for the ground state, providing a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is built upon the energy functional to obtain the ground state Wigner function representing the entire many-body system. Numerically, a parallelizable algorithm is developed using the operator splitting method and the Fourier spectral collocation method, whose numerical complexity of a single iteration is [Formula: see text]. Numerical experiments demonstrate the anticipated accuracy, encompassing the one-dimensional system with up to [Formula: see text] particles and the three-dimensional system with defect, showcasing the potential of our approach to large-scale simulations and computations of systems with defect. To the best of our knowledge, this is the first deterministic method for calculating the ground state Wigner function of the three-dimensional many-body system.