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We propose a method to study the second-order critical lines of classicalspin-$S$ Ising models on two-dimensional lattices in a crystal or splittingfield, using an exact expression for the bare mass of the underlying fieldtheory. Introducing a set of anticommuting variables to represent the partitionfunction, we derive an exact and compact expression for the bare mass of themodel including all local multi-fermions interactions. By extension of theIsing and Blume-Capel models, we extract the free energy singularities in thelow momentum limit corresponding to a vanishing bare mass. The loci of thesesingularities define the critical lines depending on the spin S, in goodagreement with previous numerical estimations. This scheme appears to begeneral enough to be applied in a variety of classical Hamiltonians.

Second-order critical lines of spin-S Ising models in a splitting field with Grassmann techniques