Critical behaviour of the extended-ballistic transition for pulled self-avoiding walks

In order to study long chain polymers many lattice models accommodate a pulling force applied to a particular part of the chain, often a free endpoint. This is in addition to well-studied features such as energetic interaction between the lattice polymer and a surface. However, the critical behaviour of the pulling force alone is less well studied, such as characterizing the nature of the phase transition and particularly the values of the associated exponents. We investigate a simple model of lattice polymers subject to forced extension, namely self-avoiding walks (SAWs) on the square and simple cubic lattices with one endpoint attached to an impermeable surface and a force applied to the other endpoint acting perpendicular to the surface. In the thermodynamic limit the system undergoes a transition to a ballistic phase as the force is varied and it is known that this transition occurs whenever the magnitude of the force is positive, i.e. $f>f_\text{c}=0$. Using well established scaling arguments we show that the crossover exponent $\phi$ for the finite-size model is identical to the well-known exponent $\nu_d$, which controls the scaling of the size of the polymer in $d$-dimensions. With extensive Monte Carlo simulations we test this conjecture and show that the value of $\phi$ is indeed consistent with the known values of $\nu_2 = 3/4$ and $\nu_3 = 0.587 597(7)$. Scaling arguments, in turn, imply the specific heat exponent $\alpha$ is $2/3$ in two dimensions and $0.29815(2)$ in three dimensions.