$OSp(n|2m)$ quantum chains with free boundaries

In this paper we investigate the spectrum of $OSp(n|2m)$ quantum spin chains with free boundary conditions. We compute the surface free energy of these models which, similar to other properties in the thermodynamic limit including the effective central charge of the underlying conformal field theory, depends on $n-2m$ only. For several models in the regime $n-2m< 2$ we have studied the finite-size properties including the subleading logarithmic corrections to scaling. As in the case of periodic boundary conditions we find the existence of a tower of states with the same conformal dimension as the identity operator. As expected the amplitudes of the corresponding logarithmic corrections differ from those found previously for the models with periodic boundary conditions. We point out however the existence of simple relations connecting such amplitudes for free and periodic boundaries. Based on our findings we formulate a conjecture on the long distance behaviour of the bulk and surface watermelon correlators.