We consider two critical semi-infinite subsystems with different critical exponents and couple them through their surfaces. The critical behavior at the interface, influenced by the critical fluctuations of the two subsystems, can be quite rich. In o
rder to examine the various possibilities, we study a system composed of two coupled Ashkin-Teller models with different four-spin couplings epsilon, on the two sides of the junction. By varying epsilon, some bulk and surface critical exponents of the two subsystems are continuously modified, which in turn changes the interface critical behavior. In particular we study the marginal situation, for which magnetic critical exponents at the interface vary continuously with the strength of the interaction parameter. The behavior expected from scaling arguments is checked by DMRG calculations.
