CONFORMAL-INVARIANCE AND CRITICAL SYSTEMS WITH MIXED BOUNDARY-CONDITIONS

Conformal-invariance methods are applied to semi-infinite two-dimensional critical systems defined on the upper half-plane with different boundary conditions a and b on the positive and negative x-axes. (The half-space with ab boundary conditions is equivalent, under a conformal mapping, to a strip with boundary conditions a and b on opposite edges.) The conformal Ward identity is derived. The n-point correlation function in the system with mixed boundary conditions follows from the same differential equations as a particular bulk (2n + 2)-point correlation function. We calculate all density profiles <phi(r)>ab determined by the condition of degeneracy at levels 2 and 3. The profiles and pair correlation functions of the magnetization and the energy density in the Ising model with +- and +f (fixed-free) boundary conditions are obtained. Corresponding results for profiles in the Q-state Potts model are given. We also derive some useful general expressions for bulk four-point correlation functions involving operators degenerate at levels 2 and 3.