The Θ points of interacting self-avoiding walks and rings on a 2D square lattice

Abstract

We propose an order parameter to locate theΘ points of interacting self-avoiding walks (ISAWs) and self-avoiding rings (SARs) on atwo dimensional square lattice. Using exact enumeration results for ISAWs offinite size we find that the order parameter as a function of temperature showsa discontinuous jump at the transition temperature. The computed transitiontemperature fluctuates with the size of the walk and appears to converge well to theΘ point as the size increases. The value for theΘ point of the linear homopolymer phase transition obtained from our methodagrees with recent high precision Monte Carlo estimates. We tested thereliability of our method for longer walk lengths by using the density of statesobtained from the recently proposed flat energy histogram method. TheΘ point is also computed for SARs on a square lattice using exact enumeration results availablein the literature. Using two other independent methods, namely the partition function zeroson the complex temperature plane and the inflection point of microcanonical entropy, theΘ point of an SAR is computed and is found to be in agreement with the result obtained withthe proposed order parameter.