Exact renormalization group exhibiting tricritical fixed point for a spin-one Ising model in one dimension

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Abstract
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.