Long-Range Order for Critical Book-Ising and Book-Percolation

In this paper, we investigate the behaviour of statistical physics models on a book with pages that are isomorphic to half-planes. We show that even for models undergoing a continuous phase transition on ℤ^2 , the phase transition becomes discontinuous as soon as the number of pages is sufficiently large. In particular, we prove that the Ising model on a three pages book has a discontinuous phase transition (if one allows oneself to consider large coupling constants along the line on which pages are glued). Our work confirms predictions in theoretical physics which relied on renormalization group, conformal field theory and numerics (Cardy in J Phys A Math Gen 24(22):L131, 1991; Iglói et al. in J Phys A Math Gen 24(17):L1031, 1991; Stéphan et al. in Phys Rev B 82(12):125455, 2010) some of which were motivated by the analysis of the Renyi entropy of certain quantum spin systems.