and find that the phase transition is of the second order with the mean field critical exponent.

The Chern-Simons term modifies the dispersion relation in such a way that the density of states per unit energy diverges at some non-zero momenta, causing the Van Hove singularity. Moreover, this happens even above the phase transition temperature. It suggests that quantum corrections to the phase transition can be significant. We argue that the order of the phase transition is changed from the second to the first due to quantum effects in the bulk.

The rest of the paper is organized as follows. In section 2, as a warm-up exercise, we will discuss non-linear solutions to the Maxwell-Chern-Simons theory in the 5-dimensional Minkowski space. In section 3, we turn to the theory in the full black hole geometry and construct non-linear solutions in the limit where the Chern-Simons coupling is large. We find that the phase transition is of the second order with the mean field exponent. In section 4, we discuss quantum corrections to the phase transition and argue the order of the phase transition is changed. We evaluate the linear response of the system in section 5. In appendix A, we 