We study the -symmetric model with the nearest neighbour interaction between molecular dipole of five spin directions i.e. Q=5 which called as the -symmetric model on a triangular lattice. We investigate the zeros of partition function and the relationship to the phase transition. Initially, the model is defined on a triangular lattice graph with the nearest neighbour interaction. The partition function is then computed using a transfer matrix approach. We analyse the system by computing the zeros of the polynomial partition function using the Newton-Raphson method and then plot the zeros in a complex plane. For this lattice, the result shows that for specific type of energy level there are multiple line curves approaching real axis in the complex plane. The equation of the specific heat is produced and then plotted for comparison. Motivated from the work by Martin (1991) on models on square lattice, we extend the previous study to different lattice type that is triangular lattice.

There is a study on a square lattice that can predict the existence of multiple phase transitions on a complex plane. We extend the study on the different types of ZQ-symmetric model and different lattices in order to provide more evidence to the existence of multiple phase transitions. We focus on the ZQ-symmetric model with the nearest neighbour interaction on the six spin directions between molecular dipole, i.e. Q = 6 on a triangular lattice. Mainly, the model is defined on the triangular lattice graph with the nearest neighbour interaction. By using the transfer matrix approach, the partition functions are computed for increasing lattice sizes. The roots of polynomial partition function are also computed and plotted in the complex Argand plane. The specific heat equation is used for further comparison. The result supports the existence of the multiple phase transitions by the emergence of the multiple line curves in the locus of zeros distribution for specific type of energy level.