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The $\Xi$ single-particle potential obtained in nuclear matter with thenext-to-leading order baryon-baryon interactions in chiral effective fieldtheory is applied to finite nuclei by an improved local-density approximationmethod. As a premise, phase shifts of $\Xi N$ elastic scattering and theresults of Faddeev calculations for the $\Xi NN$ bound state problem arepresented to show the properties of the $\Xi N$ interactions in the presentparametrization. First, the $\Xi$ states in $^{14}$N are revisited because ofthe recent experimental progress, including the discussion on the $\Xi N$spin-orbit interaction that is relevant to the location of the $p$-state. Thenthe $\Xi$ levels in $^{56}$Fe are calculated. In particular, the level shiftwhich is expected to be measured experimentally in the near future ispredicted. The smallness of the imaginary part of the $\Xi$ single-particlepotential is explicitly demonstrated.

$\Xi$ hyper-nuclear states predicted by NLO chiral baryon-baryon interactions