Edge critical behaviour of the two-dimensional tri-critical Ising model

Using previous results from boundary conformal held theory and integrability, a phase diagram is derived for the two-dimensional Ising model at its bulk tri-critical point as a function of boundary magnetic field and boundary spin-coupling constant. A boundary tri-critical point separates phases where the spins on the boundary are ordered or disordered. In the latter range of coupling constant, there is a non-zero critical field where the magnetization is singular. In the former range, as the temperature is lowered, the boundary undergoes a first-order transition while the bulk simultaneously undergoes a second-order transition.