  We explore a novel reentrant phase transition of four-dimensional
Born-Infeld-dilaton black hole in which the first order phase transition modify
into a zeroth order phase transition below the critical point. Working in the
extended phase space with regarding the cosmological constant as a pressure, we
study the reentrant behavior of phase transition in the canonical ensemble. We
show that these black holes enjoy a zeroth order intermediate-small black hole
phase transition as well as a first order phase transition between small and
large black holes for a narrow range of temperatures and pressures. We also
find that the standard first order small-large black hole phase transition can
modify into a zeroth order type. This zeroth order phase transition stands
between the critical point and the first order phase transition region. We
discuss the significant effect of the scalar field (dilaton) on the mentioned
interesting treatment.
