Exact theta point and exponents for two models of polymer chains in two dimensions.

Abstract

The collapse transition of a self-avoiding walk (SAW) on a two-dimensional lattice with directed bonds, the Manhattan lattice, is shown to occur at temperature ${T}_{\ensuremath{\theta}}$=2\ensuremath{\varepsilon}/ln2, where \ensuremath{\varepsilon} is the attractive energy between nearest-neighbor pairs of monomers. The exact tricritical exponents are ${\ensuremath{\nu}}_{t}$=(4/7 and ${\ensuremath{\gamma}}_{t}$=(6/7. The latter result differs from the value for undirected two-dimensional lattices ${\ensuremath{\gamma}}_{t}$=(8/7 because self-trapping configurations do not occur on the Manhattan lattice. The exact tricritical temperature and exponents for a constrained self-avoiding trail (SAT) on the square lattice are obtained by mapping the problem onto the self-avoiding walk on the Manhattan lattice. The mapping also shows that these SAT and SAW collapse transitions are in the same universality class. Finally, it is argued that the kinetic self-avoiding trail must be compact on the square lattice.