Zimm–Bragg model

A homogeneous Zimm-Bragg chain assigns propagation weight $s>0$ to each helical unit and initiation factor $\sigma >0$ to each helical run. With states ordered as coil and helix, one possible transfer matrix for a classical spin chain isIts largest eigenvalue is ${\lambda }_{+}=(1+s+\sqrt{(s-1{)}^2+4s\sigma })/2$. In the long-chain limit the partition function grows as ${\lambda }_{+}^N$, so logarithmic differentiation gives the helical fraction $f_h=s\partial \ln {\lambda }_{+}/\partial s$, equal to the displayed expression. If $Q=(s-1{)}^2+4s\sigma$, thenFor $\sigma =1$, $f_h=s/(1+s)$ and units are noncooperative. A small initiation factor suppresses boundaries and sharpens the crossover around $s=1$, with width of order $\sqrt{\sigma }$. For every fixed $\sigma >0$ the long-chain expression is smooth; a discontinuous step appears only in the singular $\sigma \downarrow 0$ limit. ^[ref]