On the Singularity of the Free Energy at First Order Phase Transition
  At first order phase transition the free energy does not have an analytic continuation in the thermodynamical variable, which is conjugate to an order parameter for the transition. This result is proved at low temperature for lattice models with finite range interaction and two periodic ground-states, under the only condition that they verify Peierls condition. 
S. Friedli, Ch.-Ed. Pfister