Non-perturbative tricritical exponents of trails. II. Exact enumerations on square and simple cubic lattices

Universality class of the special adsorption point of two-dimensional lattice polymers

In recent work [Rodrigues et al., Phys. Rev. E 100, 022121 (2019)10.1103/PhysRevE.100.022121], evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a nonuniversal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents ϕ^{(s)} and 1/δ^{(s)} were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS), on which the trails are forbidden from having two consecutive steps, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar 1/δ^{(s)} exponents at the SAP, namely, 1/δ^{(s)}≈0.44, whereas 1/δ^{(s)}≈0.34 in the HS case. Intriguingly, while ϕ^{(s)}≈1/δ^{(s)} is found for the DS and HS scenarios, as expected, in the mHS case ϕ^{(s)} is about 10% smaller than 1/δ^{(s)}. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.

Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model

The complex zeros of the canonical (fixed walk-length) partition function are calculated for both the self-avoiding trails model and the vertex-interacting self-avoiding walk model, both in bulk and in the presence of an attractive surface. The finite-size behavior of the zeros is used to estimate the location of phase transitions: the collapse transition in the bulk and the adsorption transition in the presence of a surface. The bulk and surface cross-over exponents, ϕ and ϕ S , are estimated from the scaling behavior of the leading partition function zeros.

Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

We present results from extensive Monte Carlo simulations of polymer models where each lattice site can be visited by up to K monomers and no restriction is imposed on the number of bonds on each lattice edge. These multiple monomer per site (MMS) models are investigated on the square and cubic lattices, for K=2 and 3, by associating Boltzmann weights ω_{0}=1, ω_{1}=e^{β_{1}}, and ω_{2}=e^{β_{2}} to sites visited by 1, 2, and 3 monomers, respectively. Two versions of the MMS models are considered for which immediate reversals of the walks are allowed (RA) or forbidden (RF). In contrast to previous simulations of these models, we find the same thermodynamic behavior for both RA and RF versions. In three dimensions, the phase diagrams, in space β_{2}×β_{1}, are featured by coil and globule phases separated by a line of Θ points, as thoroughly demonstrated by the metric ν_{t}, crossover ϕ_{t}, and entropic γ_{t} exponents. The existence of the Θ lines is also confirmed by the second virial coefficient. This shows that no discontinuous collapse transition exists in these models, in contrast to previous claims based on a weak bimodality observed in some distributions, which indeed exists in a narrow region very close to the Θ line when β_{1}<0. Interestingly, in two dimensions, only a crossover is found between the coil and globule phases.

Nature of the collapse transition in interacting self-avoiding trails

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 ω_{i} with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q=4 and K=2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs 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.

Generalized interacting self-avoiding trails on the square lattice: phase diagram and critical behavior

A generalized model for interacting self-avoiding trails on a square lattice is presented and studied using numerical transfer matrix methods. The model differentiates between on-site double visits corresponding to collisions, and crossings. Rigidity is also included in the model. The model includes the Nienhuis O(n=0) model and the interacting self-avoiding trail model as special cases. It is shown that the generic type of collapse found is the same as in the pure interacting self-avoiding trail model.