Simple model for the DNA denaturation transition

We study pairs of interacting self-avoiding walks $\{{\mathit{\omega }}^1,{\mathit{\omega }}^2\}$ on the $3d$ simple cubic lattice. They have a common origin ${\mathit{\omega }}_0^1={\mathit{\omega }}_0^2,$ and are allowed to overlap only at the same monomer position along the chain: ${\mathit{\omega }}_i^1\ne {\mathit{\omega }}_j^2$ for $i\ne j,$ while ${\mathit{\omega }}_i^1={\mathit{\omega }}_i^2$ is allowed. The latter overlaps are indeed favored by an energetic gain ε. This is inspired by a model introduced long ago by Poland and Sheraga [J. Chem. Phys. 45, 1464 (1966)] for the denaturation transition in DNA where, however, self avoidance was not fully taken into account. For both models, there exists a temperature $T_m$ above which the entropic advantage to open up overcomes the energy gained by forming tightly bound two-stranded structures. Numerical simulations of our model indicate that the transition is of first order (the energy density is discontinuous), but the analog of the surface tension vanishes and the scaling laws near the transition point are exactly those of a second-order transition with crossover exponent $\phi =1.\mathrm{}$ Numerical and exact analytic results show that the transition is second order in modified models where the self-avoidance is partially or completely neglected.