Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and ‘modified’ energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods.
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