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

TL;DR

This paper investigates conformal defects in free fermions, classifies boundary conditions, and supports a defect $C$-theorem through free energy comparisons during RG flows.

Contribution

It classifies boundary conditions for conformal defects in free fermions and provides evidence for a defect $C$-theorem via free energy analysis.

Findings

Dirichlet boundary conditions always exist.

Neumann boundary conditions exist only for two-codimensional defects.

RG flow from Neumann to Dirichlet reduces free energy, supporting the $C$-theorem.

Abstract

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.