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We consider the $varphi_{1,3}$ off-critical perturbation ${cal M}(m,m;t)$ of the general non-unitary minimal models where $2le mle m$ and $m, m$ are coprime and $t$ measures the departure from criticality corresponding to the $varphi_{1,3}$ integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to $mathbb{Z}_n$ parfermions with $n=m-2$. Using an elliptic Yang-Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the $A_{m-1}$ Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.
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