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 ${\omega }_i$ 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.

Figure 1

Three walks on the square lattice (from left to right): a SAW, a SAT, and a MMS configuration. The MMS walk and trail shown here visit each lattice site at most twice. For trails on the square lattice, this is the upper limit for the number of visits of a site.

Figure 2

Contribution to the partition function of the model on a Cayley tree with $q=4$ and 3 generations. The weight of this contribution will be ${\omega }_1^9{\omega }_2^2$. The end points of the walks are placed on the surface of the Cayley tree.

Figure 3

Phase diagram for $q=4$ with $K=2$. The solid vertical (red) and slanted (blue) lines are the P-NP and P-DP critical lines, respectively. The NP and DP phases coexist on the dashed (black) line. The (black) circle is the bicritical point.

Figure 4

Phase diagram for $q=6$ with $K=2$. The solid (vertical) and dashed lines are the critical and coexistence lines, respectively. The circle is the tricritical point.

Figure 5

Phase diagrams for $q=6$ with $K=3$ and several values of ${\omega }_3<1/15$. The solid (vertical) line represents the critical plane separating the NP and polymerized phases. The circle indicates a tricritical line (TCL). The dashed lines represent the NP-P coexistence surface for (from top to bottom) ${\omega }_3=0$, 0.02, 0.04, and 0.06.

Figure 6

Phase diagrams for $q=6$ with $K=3$ and several ${\omega }_2<1/15$. The solid vertical (red) line represents the critical P-NP plane. The dashed horizontal (black) line indicates the coexistence surface of NP and DP phases. The dashed (blue) lines are P-DP coexistence lines for (from bottom to top) ${\omega }_2=0.0$, 0.02, 0.04, and 0.06. The (violet) circle represents the NP-P critical end-point line.

Figure 7

Phase diagrams for $q=6$ with $K=3$ and (a) several values of ${\omega }_1<1/5$, (b) ${\omega }_1=1/5$, and (c) ${\omega }_1=0.25$. In (a) the solid (slanted, blue) and dashed (horizontal, black) lines represent the critical P-DP and the coexistence NP-DP planes, respectively. The dashed (red) lines constitute the P-NP coexistence surface for (from right to left) ${\omega }_1=0$, 0.08, 0.16, and 0.199. In (b), at the multicritical point (black square), the P-NP critical end-point line (horizontal, violet), the tricritical P-NP line (vertical, red), and the critical P-DP line (slanted, blue) meet. In (c), at the tricritical point (circle), the critical (solid) and coexistence (dashed) P-DP lines meet.

Figure 8

Phase diagram for $q=6$ with $K=3$ and $\omega ={\omega }_2={\omega }_3$. At the multicritical point (black square) the P-DP critical end-point line (horizontal, green), the critical P-NP line (vertical, red), and the P-DP tricritical line (slanted, blue) meet.

Figure 9

Sketch of the three-dimensional phase diagram for $q=6$ with $K=3$. Colors and line type follow the same definitions as in the previous figures.

Figure 10

Contribution to the partition function of the model on a Husimi tree with $q=4$ and 3 generations of squares. The weight of this contribution will be ${\omega }_1^8{\omega }_2^2$. The end points of the walks are placed on the surface of the tree.

Figure 11

Definition of the root sites on the Husimi lattice. The thick (red) lines indicate incident bonds. The difference between $g_2$ and $g_3$ is that in the last (first) one the incident bonds are (not) connected.

Figure 12

Phase diagram for the ISAT model on a Husimi lattice built with squares and coordination $q=4$. The solid vertical (red) and slanted (blue) lines are the P1-NP and P1-P2 critical lines, respectively. The NP and P2 phases coexist on the dashed (black) line. The (black) circle is the bicritical point.

Figure 13

Canonical phase diagrams for (a) $q=4$ with $K=2$ and (b) $q=6$ with $K=3$. In (b), the vertical (red), horizontal (violet), and slanted (green) lines are the P-NP TCL, P-NP CEP line, and P-DP CEP line, respectively, which meet at the multicritical point (black square).