Fluxification and scalarization of the conifold black holes

We study thermal states of the strongly coupled Klebanov-Witten ${\cal N}=1$ superconformal gauge theory with R-symmetry chemical potential. The theory has two distinct dimension $\Delta=3$ chiral primary operators that develop a condensate at the same critical value of the chemical potential. One instability is associated with the spontaneous fluxification (the appearance of the 3-form flux) of the holographic dual black hole, while the other one is associated with the spontaneous scalarization (the activation of the deformation mode of the conifold). Different phases dominate in grand canonical and microcanonical ensembles.