  We investigate the evolution of phase transition of the classical fully
compact dimer model on the bipartite square lattice with second nearest bonds
at finite temperatures. We use the numeric Monte Carlo method with the
directed-loop algorithm to simulate the model. Our results show that the order
of the phase transition depends on the fugacity of the second nearest bonds. We
find that the phase transition reduces from the Kosterlitz-Thouless transition
to unconventional high-order phase transitions which feature the coexistence of
properties Kosterliz-Thouless transition and the first-order phases transition
simultaneously. As the fugacity increases further, phase transition evolves to
the first-order phase transition. In addition, our results of dimer-dimer
correlation functions and their corresponding structure factor functions
computed by us show the evolution of decay correlation for different fugacity.
