Unconventional critical behavior of polymers at sticky boundaries

TL;DR

This paper explores how modifying boundary interactions and local hopping parameters in polymer models can lead to unconventional phase transition orders, including third-order transitions, challenging classical second-order transition expectations.

Contribution

It introduces two models where the order of the polymer adsorption transition can be tuned by boundary scaling and local hopping amplitudes, revealing new critical behaviors.

Findings

Phase transition order can be altered by boundary scaling.

Polymer models can exhibit third-order phase transitions.

Adjusting local hopping amplitudes influences transition nature.

Abstract

We discuss the generalization of a classical problem involving an $N$-step ideal polymer adsorption at a sticky boundary (potential well of depth $U$). It is known that as $N$ approaches infinity, the path undergoes a 2nd-order localization transition at a certain value of $U_{\text{tr}}$. By considering the random walk on a half-line with a sticky boundary (Model I), we demonstrate that the order of the phase transition can be altered by adjusting the scaling of the first return probability to the boundary. Additionally, we present a model of a random path on a discrete 1D lattice with non-uniform local hopping amplitudes and a potential well at the boundary (Model II). We illustrate that one can tailor such amplitudes so that the polymer undergoes a 3rd-order phase transition.