We study the fractal properties of single-particle eigenmodes of an entanglement Hamiltonian in free-fermion models. One of these modes that has the highest entanglement information and is thus called the maximally entangled mode (MEM) is especially considered. In free-fermion models with Anderson localization, the fractality of the MEM is obtained numerically and compared with the fractality of the Hamiltonian eigenmode at the Fermi level. We show that both eigenmodes have similar fractal properties: both have the same single-fractal dimension in the delocalized phase which equals the dimension of the system, and both show multifractality at the phase transition point. Therefore, we conclude that fractal behavior of the MEM, in addition to the fractal behavior of the Hamiltonian eigenmode, can be used as a quantum phase transition characterization.
