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

We consider a model of semiflexible interacting self-avoiding trails (sISATs) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting self-avoiding walks (SASAWs) to investigate the collapse transition of polymers, with the attractive interactions being on site as opposed to nearest-neighbor interactions in SASAWs. The grand-canonical version of the sISAT model is solved on a four-coordinated Bethe lattice, and four phases appear: non-polymerized (NP), regular polymerized (P), dense polymerized (DP), and anisotropic nematic (AN), the last one present in the phase diagram only for sufficiently stiff chains. The last two phases are dense, in the sense that all lattice sites are visited once in the AN phase and twice in the DP phase. In general, critical NP-P and DP-P transition surfaces meet with a NP-DP coexistence surface at a line of bicritical points. The region in which the AN phase is stable is limited by a discontinuous critical transition to the P phase, and we study this somewhat unusual transition in some detail. In the limit of rods, where the chains are totally rigid, the P phase is absent and the three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point, which is the endpoint of the bicritical line.

Figure 1

Illustration of a self-avoiding trail on a square lattice. Collisions and crossings are indicated by ${\tau }_c$ and ${\tau }_x$, respectively, while bends at sites with two incoming bonds are indicated by $\omega$.

Figure 2

A configuration of three chains placed on a Cayley tree with four generations of sites. The statistical weight of this configuration is $z^{16}{\omega }^3{\tau }_c{\tau }_x$.

Figure 3

Phase diagrams for (a) $\omega =0.75$, (b) $\omega =0.25$, (c) $\omega =0.1$, and (d) $\omega =0.0$. Regular continuous transitions are shown as solid lines, discontinuous transitions are represented as dashed lines, and the dash-dotted line corresponds to the critical discontinuous transitions. The dot in panel (d) is the endpoint of the bicritical line, where the three discontinuous transition lines meet.

Figure 4

Total densities of bonds as functions of $z$ for $\tau =0.5$ and (a) $\omega =0.25$, (b) $\omega =0.1$, and (c) $\omega =0.0$.

Figure 5

Susceptibility ${\chi }_{\text{N}}$ in the P phase as a function of the bond activity $z$ for $\tau =0.1$, close to the P-AN transition. The curves, from left to right, correspond to $\omega =0.15$ (blue), $\omega =0.20$ (red), and $\omega =0.25$ (black). Dashed lines indicate the corresponding critical activities.

Figure 6

Bicritical lines in terms of (on-site) monomer-monomer interaction ${\tau }^{*}$ against $\omega$ for ISAT and VISAW models.

Figure 7

A possible configuration of one vertical and two horizontal rods on a strip with $L_x=5$ and $L_y=3$. The element of the transfer matrix corresponding to the two successive $(1,0,1)$ configurations of horizontal edges indicated by the dashed lines is $z_1^2{z}_2^3{\tau }_x^2$.