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