Nature of the collapse transition in interacting self-avoiding trails

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 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.

Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice

“…As discussed above, for ω ≥ 1/2 the AN phase is not present in the phase diagrams, see an example in Fig. 3(a) for ω = 0.75, which is qualitatively similar to the one obtained in the flexible case (ω = 1) [17]. For ω < 1/2, the thermodynamic behavior is still the same, except for the presence of the AN phase, as well as the critical discontinuous P-AN surface.…”

“…Semi-flexible trails on a Bethe lattice with coordination number equal to 4 show a very rich phase diagram in the parameter space defined by the activity of a bond (z), the statistical weights of a crossing and a collision (τ c and τ x ), and the statistical weight of an elementary bend in the trail (ω). For sufficiently flexible chains (ω > 1/2) the phase diagrams are qualitatively similar to the one found in the flexible case (ω = 1), studied in [17], with non-polymerized (NP), regular polymerized (P) and dense polymerized (DP) phases meeting at a bicritical point. When the Boltzmann factor of bends is smaller than 1/2, an additional polymerized phase appears inside the P phase.…”

“…This is done here for Bethe lattice, with different statistical weights associated to crossings and collisions, so that semi-flexible VISAW and semi-flexible ISAT are obtained as particular cases of our model. It is shown that the inclusion of semi-flexibility does not change the nature of the collapse transition when compared with the flexible ISAT model studied before [17], but an additional polymerized phase appears inside the regular polymerized phase, which is both dense and nematic, since all lattice sites are visited and all bonds are in the same direction. For SAW, the semi-flexible and the self-attracting cases were studied on the Bethe lattice [20].…”

“…It is, therefore, interesting to study these phase diagrams with approximate analytic tools, among them solutions on hierarchical lattices such as the Bethe and the Husimi lattice. Indeed, the ISAT model was recently studied on a four-coordinated Husimi lattice built with squares and on a four and sixcoordinated Bethe lattice [17]. Rich phase diagrams were found in these studies, with the coil-globule transition for four-coordinated cases (which are mean-field approximations for the square lattice) being associated with a bicritical point.…”

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

“…However, recent evidences against this exists coming from numerical [20,21] and field theory [10] works. Controversies exist also on the ISAT collapse transition [22][23][24][25][26][27][28], which have motivated several recent works on this model and generalizations of it [29][30][31][32][33][34][35]. Noteworthy among these works is the field theory by Nahum et al [30] showing that the ISAT collapse transition in 2D is multicritical with infinite order.…”