We study the ground state of a two-component dipolar Fermi gas in a spherically symmetric harmonic trap at zero temperature. We obtain the ground state of the system with equal but opposite values of the magnetic moment on the basis of the Thomas-Fermi-von Weizsäcker approximation. It is found that two types of phase-separated states which break the spherical symmetry emerge across the phase transition between a spin-symmetric phase and a phase-separation phase. One is an axially symmetric phase-separated state and the other is a one-dimensional alternating-spin layered state. We also study a Fermi-Fermi mixture of polarized Dy and Er atoms and show that a structural transition of phase separation can be observed in current experiments by varying the relative number of particles.
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