Critical gaps of first-order phase transition in infinitely long Ising   cylinders with antiperiodically joined circumference
  Based on the analytic expression of free energy for infinitely long Ising strip with finite width joined antiperiodically on a variety of planar lattices, we show the existence of first-order phase transition at the critical point of Ising transition. The critical gaps of the transition are also calculated analytically by measuring the discontinuities in the internal energy and the specific heat. 
Tsong-Ming Liaw, Ming-Chang Huang, Simon C. Lin, and Yu-Pin Luo