Free energy and defect $C$-theorem in free fermion

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