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# Lattice polymers with two competing collapse interactions

Abstract

We study a generalised model of self-avoiding trails, containing two different types of interaction (nearest-neighbour contacts and multiply visited sites), using computer simulations. This model contains various previously-studied models as special cases.

We find that the strong collapse transition induced by multiply-visited sites is a singular point in the phase diagram and corresponds to a higher order multi-critical point separating a line of weak second-order transitions from a line of first-order transitions.

# Introduction

\label{sec:introduction}

## The polymer collapse transition

There are various statistical mechanical models of the polymer collapse phase transition (Gennes 1979, Vanderzande 1998). However, there are two basic elements in each of the models: on the one (Gennes 1979) hand, the configurations of the polymer have some type of so-called excluded volume, that implies molecules are separate in space, and on the other hand the configurations have an attractive force between different parts of the polymer, that drives the transition. On lattices, both self-avoiding walks (SAW), where different sites of walk avoid being on the same site of the lattice, and self-avoiding trails (SAT), which are walks that can share sites though not bonds of the lattice, have been used as the configuration space for collapse models. The attractive force has been modelled both by adding energies for shared sites and also via so-called nearest-neighbour contacts, where sites adjacent on the lattice not joined by a step of the walk are given an energy. When SAW are coupled with nearest neighbour interactions the canonical ISAW model is reproduced (see (Vanderzande 1998) for a review), while when trails are coupled with site interactions the ISAT model (Malakis 1976) is reproduced. Despite what one might expect from the principle of universality the collapse of these models appears to behave differently (Shapir 1984, Owczarek 1995, Owczarek 2007).

A recent study (Bedini 2013) considered self-avoiding trails interacting via nearest-neighbour contacts (INNSAT) as a hybrid of the two models. Evidence from computer simulations showed that the collapse transition in INNSAT is different from the collapse transition in ISAT, which is a strong second order transition, but similar to ISAW which is predicted to be a weak second order transition where the specific heat converges at the transition. In ISAW collapse one needs to consider the third derivative of the free energy to see a divergent quantity and then only weakly divergent. It was also found that the low-temperature phase of the two trail-collapse models differ substantially: the phase associated with multiply visited site interactions is fully dense in the thermodynamic limit (as shown in (Bedini 2013a)). The low-temperature phase associated with nearest-neighbour contacts wasn’t fully dense as is believed to be the case for interacting self-avoiding walks (Owczarek 1993).

## Magnetic Systems

The properties of lattice polymers are also related to those of magnetic systems near their critical point (Vanderzande 1998). More precisely, self-avoiding walks configurations appear as the diagrams of the high-temperature expansion of an $$O(n)$$ magnetic system when taking the formal limit of zero components ($$n \to 0$$), and their scaling exponents can be obtained from the $$O(n)$$ critical point. In this mapping the collapse transition corresponds to a tri-critical point of the magnetic system and one would hope to obtain the critical exponents for the polymer collapse transition from the ones of this tri-critical point.

Various authors (Nienhuis 1982, Nienhuis 1984, Blote 1989, Guo 2006, Nienhuis 2008) have studied critical and tri-critical $$O(n)$$ spin systems. For a special choice of the model on the honeycomb lattice, exact results were obtained in (Nienhuis 1982) for two cases: a critical point and a special point governing the low-temperature phase. When $$n \to 0$$ these two cases become the dilute and dense polymer phase. The dilute and dense phases were also found along two branches of a square-lattice $$O(n)$$ model (Nienhuis 1984, Blote 1989) together with two different branches describing the critical behaviour that occurs when $$O(n)$$ and Ising degrees of freedom on the square lattice display a joint critical point.

On the other hand, Duplantier and Saleur in 1987 (Duplantier 1987) realised that, on the honeycomb lattice, an self-interaction for SAW could be obtained by introducing vacancies, hexagonal faces that the SAW is not allowed to touch. Using this observation they could obtain a set of critical exponents for the polymer collapse transition which have been subsequently found to correctly describe the collapse in the ISAW model (see the extensive list of references in (Caracciolo 2011) for example). We will refer to the universality class of this critical point as the ‘$$\theta$$-point’. An exact description has now been proposed (Guo 2006, Nienhuis 2008) for the tri-critical $$O(n)$$ model in two dimensions as a function of $$n$$.

When it comes to ISAT the scenario is much less clear, in particular it not obvious how the change of topology caused by the presence of crossings affects the above picture. The description in terms of height model and Coulomb Gas allows one to consider the presence of crossings only as a perturbation (see (Jacobsen 2009) for a review of these methods). The exponent associated to loop crossings is the same as that of cubic symmetry breaking, which is known to be irrelevant in the critical $$O(n)$$ phase, but it has been observed (Jacobsen 2003) that this is not true in the low-temperature phase, where the introduction of crossings is a relevant operator which leads to a different universality class. This is generically referred to as the Goldstone phase and it is believed (Jacobsen 2003, Nienhuis 2008) to be described by the intersecting loop model proposed in (Martins 1998, Martins 1998) and since called the Brauer model (Gier 2005).

The relevance of crossings at the collapse point is not clear. While the cubic perturbation is still believed to be relevant at the tri-critical $$O(n)$$ point (Nienhuis 2008), a recent numerical study (Bedini 2013) seems to indicate that the Duplantier and Saleur universality class is stable in the presence of crossings, at least with respect to the cross-over and length-scale exponents.

Very recently Nahum et al. (Nahum 2013) published a study of loop models with crossings. Their analysis is based on a replica limit of the $$\sigma$$ model on real projective space $$\mathbb{RP}^{n-1}$$. They give a field theoretic description of the ISAT which explains the phase diagram found numerically in (Foster 2009) and suggests that the ISAT collapse transition is an infinite-order multi-critical point.