Bethe Approximation for a Semi-flexible Polymer Chain

We present a Bethe approximation to study lattice models of linear polymers. The approach is variational in nature and based on the cluster variation method (CVM). We focus on a model with $(i)$ a nearest neighbor attractive energy $\epsilon_v$ between pair of non--bonded monomers, $(ii)$ a bending energy $\epsilon_h$ for each pair of successive chain segments which are not collinear. We determine the phase diagram of the system as a function of the reduced temperature $t=\frac{T}{\epsilon_v}$ and of the parameter $x=\frac{\epsilon_h}{\epsilon_v}$. We find two different qualitative behaviors, on varying $t$. For small values of $x$ the system undergoes a $\theta$ collapse from an extended coil to a compact globule; subsequently, on decreasing further $t$, there is a first order transition to an anisotropic phase, characterized by global orientational order. For sufficiently large values of $x$, instead, there is directly a first order transition from the coil to the orientational ordered phase. Our results are in good agreement with previous Monte Carlo simulations and contradict in some aspects mean--field theory. In the limit of Hamiltonian walks, our approximation recovers results of the Flory-Huggins theory for polymer melting.