Nature of the collapse transition in interacting self-avoiding trails

TL;DR

This paper provides exact solutions for the phase behavior of the interacting self-avoiding trail model on Bethe and Husimi lattices, revealing complex phase diagrams and the nature of the collapse transition.

Contribution

It offers the first exact grand-canonical solutions for the ISAT model on these lattices, elucidating the collapse transition's nature and phase diagram complexity.

Findings

Collapse transition is bicritical on Bethe lattice with q=4 and K=2.

Dense polymerized phase is associated with the collapsed state.

Complex phase diagrams with multiple critical and coexistence lines are found for q=6, K=3.

Abstract

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination $q$ and on a Husimi lattice built with squares and coordination $q=4$. The exact grand-canonical solutions of the model are obtained, considering that up to $K$ monomers can be placed on a site and associating a weight $\omega_i$ for a $i$-fold visited site. Very rich phase diagrams are found with non-polymerized (NP), regular polymerized (P) and dense polymerized (DP) phases separated by lines (or surfaces) of continuous and discontinuous transitions. For Bethe lattice with $q=4$ and $K=2$, the collapse transition is identified with a bicritical point and the collapsed phase is associated to the dense polymerized phase (solid-like) instead of the regular polymerized phase (liquid-like). A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISAT's and for interacting self-avoiding walks on the square lattice. For $q=6$ and $K=3$ (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.