  Higher-order topological insulators in two dimensions have states that
localize at their corners, called corner states. In this paper, we reveal
characteristics of the penetration depth of their corner states by using the
Benalcazar-Bernevig-Hughes model. First, we show that when we change the energy
of the corner states toward the end of the edge gap by adding an on-site
potential to the corner site, the penetration depth along the edge diverges
toward infinity while the penetration depth into the bulk remaining finite. We
analytically derive the corner-state wavefunction in a form of elliptic
integrals, which reproduces this anisotropic behavior of corner states. This
means that corner states have two kinds of penetration depths, and they behave
differently. At last, we show that hybridizations between corner states are
governed by the penetration depth through interference between the corner
states. It is because the corner states almost do not interfere with edge
states or bulk states.
