  We study an explicit formula for the thermodynamic potential of the AdS
dyonic black brane solution with an axio-dilaton hair, which was discovered in
an extension of the $(3+1)$d Gubser-Rocha model enjoying S-duality. From the
thermodynamic potential, we can compute the magnetization and the magnetic
susceptibilities of the dyonic solution. The result of the magnetization is
negative implying that the system is diamagnetic. Subsequently, we consider a
specific neutral limit of the dyonic solution. In this setup, we find that the
system exhibits spontaneous breaking of a global $U(1)$ symmetry. The order
parameter is given by a complex operator which is dual to the axio-dilaton
field in the bulk. Interestingly, the system has a finite Hall conductivity
even in the absence of the external magnetic field, and it is related to the
phase of the complex operator.
