  Defects are common in physical systems with boundaries, impurities or
extensive measurements. The interaction between bulk and defect can lead to
rich physical phenomena. Defects in gapless phases of matter with conformal
symmetry usually flow to a defect conformal field theory (dCFT). Understanding
the universal properties of dCFTs is a challenging task. In this paper, we
propose a computational strategy applicable to a line defect in arbitrary
dimensions. Our main assumption is that the defect has a UV description in
terms of a local modification of the Hamiltonian so that we can compute the
overlap between low-energy eigenstates of a system with or without the defect
insertion. We argue that these overlaps contain a wealth of conformal data,
including the $g$-function, which is an RG monotonic quantity that
distinguishes different dCFTs, the scaling dimensions of defect creation
operators $\Delta^{+0}_\alpha$ and changing operators $\Delta^{+-}_\alpha$ that
live on the intersection of different types of line defects, and various OPE
coefficients. We apply this method to the fuzzy sphere regularization of 3D
CFTs and study the magnetic line defect of the 3D Ising CFT. Using exact
diagonalization and DMRG, we report the non-perturbative results
$g=0.602(2),\Delta^{+0}_0=0.108(5)$ and $\Delta^{+-}_0=0.84(5)$ for the first
time. We also obtain other OPE coefficients and scaling dimensions. Our results
have significant physical implications. For example, they constrain the
possible occurrence of spontaneous symmetry breaking at line defects of the 3D
Ising CFT. Our method can be potentially applied to various other dCFTs, such
as plane defects and Wilson lines in gauge theories.
