Ising model on self-avoiding walk chains

Abstract

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T_c) bounds, the average number of nearest-neighbours (Z_eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p_c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively, Z_eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) and p_c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values of Z eff give kT _c/ J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising model T _c's and p _c's for the same lattices, we find kT c/ J<0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent γ=1.024±0.007, for the model on SAWs on two dimensional lattices.