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Explicit R-matrices for inhomogeneous 3D chiral Potts models: Integrability and the action formulation for

We construct the exact spectral parameter dependent vertex R-matrix for the classical 3D $\mathcal{N}$-state chiral Potts models, convenient for considering the model in context of the Bethe ansatz. The R-matrix is defined on the $\mathcal{N}^4$ dimensional space $V_\mathcal{N}\otimes V_\mathcal{N}\otimes V_\mathcal{N}\otimes V_\mathcal{N}$, appropriate for consideration by means of the cube-equations defined in [14]. We present the 2D quantum spin Hamiltonians for general case and, at $\mathcal{N}=2$, a fermionic lattice action representation corresponding to 3D Ising's statistical model.