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<p>The ground state for the ultracold Fermi gas with dipole-dipole interaction is a functional minimization problem based on density functional theory (DFT). We extend the recent work on Sobolev gradient flows for the Gross-Pitaevskii eigenvalue problem [SIAM J. Numer. Anal. <b>58</b>(3), 1744–1772 (2020)], and present continuous projected Sobolev gradient flows for computing the DFT-based ground state solution of ultracold dipolar Fermi gas. We prove that the gradient flows have the properties of orthonormality preserving and energy diminishing, which are desirable for the computation of the ground state solution. Many numerical techniques for partial differential equations can be used to discretize the time-dependent projected Sobolev gradient flows, which may be an advantage of the method. We propose an efficient and accurate numerical scheme – semi-implicit Euler method in time and Fourier spectral method in space for discretizing these projected Sobolev gradient flows and use them to find the ground state of the ultracold dipolar Fermi gas numerically. Extensive numerical examples in three dimensions for the ground state are reported to demonstrate the power of the numerical methods.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月3日

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  • Xuelin Zhang,
  • Chunping Pang,
  • Hanquan Wang

摘要

The ground state for the ultracold Fermi gas with dipole-dipole interaction is a functional minimization problem based on density functional theory (DFT). We extend the recent work on Sobolev gradient flows for the Gross-Pitaevskii eigenvalue problem [SIAM J. Numer. Anal. 58(3), 1744–1772 (2020)], and present continuous projected Sobolev gradient flows for computing the DFT-based ground state solution of ultracold dipolar Fermi gas. We prove that the gradient flows have the properties of orthonormality preserving and energy diminishing, which are desirable for the computation of the ground state solution. Many numerical techniques for partial differential equations can be used to discretize the time-dependent projected Sobolev gradient flows, which may be an advantage of the method. We propose an efficient and accurate numerical scheme – semi-implicit Euler method in time and Fourier spectral method in space for discretizing these projected Sobolev gradient flows and use them to find the ground state of the ultracold dipolar Fermi gas numerically. Extensive numerical examples in three dimensions for the ground state are reported to demonstrate the power of the numerical methods.