Boundary Symmetry Breaking in CFT and the String Order Parameter

We consider the ground state of a one-dimensional critical quantum system carrying a global symmetry in the bulk, which is explicitly broken by its boundary conditions. We probe the system via a string-order parameter, showing how it detects the symmetry breaking pattern. We give a precise characterization of the mechanism depicted above in Boundary CFT, and we find a general logarithmic scaling for the order parameter. As a first example we analyze the breaking of a $U(1)$ symmetry for complex free theories induced by a boundary pairing term. Moreover, we give predictions for the breaking of $U(N)$ in free theories, arising from a boundary mixing. We test our predictions with numerical calculations for some lattice realizations of free fermionic system with boundary symmetry breaking, finding a good agreement.