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Equivalence between non-bilinear spin-$S$ Ising model and Wajnflasz model

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-$S$ (for simplicity, we called as spin-$S$ polynomial) onto spin-crossover state. The spin-$S$ polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-$S$ is given by $2(2^{2S}-1)$. As an application of this mapping, we consider a general non-bilinear spin-$S$ Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-$S$ Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-$S$ Ising model.