Free energy and defect C-theorem in free fermion

We describe a $p$-dimensional conformal defect of free Dirac fermion on $d$-dimensional flat space as boundary conditions conformally equivalent $\mathbb{H}^{p+1} \times \mathbb{S}^{d-p-1}$. classify allowed and find that the Dirichlet type always exists while Neumann condition only for two-codimensional defect. For defect, double trace deformation triggers renormalization group flow from to condition, energy at UV fixed point is larger than IR point. This provides us with further support conjectured $C$-theorem in DCFT.