In an armchair carbon nanotube pn junction the p and n regions are separated by a region of a Mott insulator, which can backscatter electrons only in pairs. We predict a quantum-critical behavior in such a pn junction. Depending on the junction's built-in electric field E, its conductance G scales either to zero or to 4e2/h as the temperature T is lowered. The two types of the G(T) dependence indicate the existence, at some special value of E, of an intermediate quantum-critical point with a finite conductance G<4e2/h. This makes the pn junction drastically different from a simple potential barrier in a Luttinger liquid.