Bethe approximation for a semiflexible 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. We focus on a model with (i) a nearest-neighbor attractive energy ${\epsilon }_v$ between a pair of nonbonded monomers, (ii) a bending energy ${\epsilon }_h$ for each pair of successive chain segments that are not collinear. We determine the phase diagram of the system as a function of the reduced temperature $t=T/{\epsilon }_v$ and of the parameter $x={\epsilon }_h/{\epsilon }_v.$ We find two different qualitative behaviors, on varying t. For small values of x the system undergoes a θ 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.