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Anomalous critical behaviour in the polymer collapse transition of three-dimensional lattice trails

Trails (bond-avoiding walks) provide an alternative lattice model of polymers to self-avoiding walks, and adding self-interaction at multiply visited sites gives a model of polymer collapse. Recently, a two-dimensional model (triangular lattice) where doubly and triply visited sites are given different weights was shown to display a rich phase diagram with first and second order collapse separated by a multi-critical point. A kinetic growth process of trails (KGT) was conjectured to map precisely to this multi-critical point. Two types of low temperature phases, globule and crystal-like, were encountered. Here, we investigate the collapse properties of a similar extended model of interacting lattice trails on the simple cubic lattice with separate weights for doubly and triply visited sites. Again we find first and second order collapse transitions dependent on the relative sizes of the doubly and triply visited energies. However we find no evidence of a low temperature crystal-like phase with only the globular phase in existence. Intriguingly, when the ratio of the energies is precisely that which separates the first order from the second-order regions anomalous finite-sized scaling appears. At the finite size location of the rounded transition clear evidence exists for a first order transition that persists in the thermodynamic limit. This location moves as the length increases, with its limit apparently at the point that maps to a KGT. However, if one fixes the temperature to sit at exactly this KGT point then only a critical point can be deduced from the data. The resolution of this apparent contradiction lies in the breaking of crossover scaling and the difference in the shift and transition width (crossover) exponents.