Quantum entropy of two-dimensional extreme charged dilaton black hole

By using Hawkingʹs treatment as well as Zaslavskiiʹs treatment respectively and the brick wall model, two different values of classical entropy and quantum entropy of scalar fields in the two-dimensional extreme charged dilaton black hole backgrounds have been obtained. A new divergent term emerges in the quantum entropy under the extreme limit for Zaslavskiiʹs treatment and its connection with the phase transition has been addressed.