Thermodynamic Insights into the Impact of Increasing Connectivity for 2D-Lattices Based on Ising Chains

The Ising model provides a fundamental setting for investigating the emergence of phase transitions from simple interacting degrees of freedom. The current characterization study serves to investigate central requirements for phase transitions in terms of connectivity, i.e., the degree of coupled interactions between interaction sites. The impact of increasing connectivity between 1D-Ising chains mapped onto 2D-lattices with free boundary conditions were studied systematically, using exact free energy calculations. Starting from a reference system of non-interacting 1D-chains, interaction bonds between chains are introduced successively until the fully connected N×N-lattice is obtained. Two distinct construction schemes are analyzed, which differ in the connectivity of the intermediate partially coupled systems. The resulting free energies of the graphs along these paths are evaluated and compared with respect to their convergence behavior as a function of system size. We find that, despite topological differences between the schemes, strikingly, they converge to the same limiting straight line for increasing N when analyzed in terms of residual free energy differences. These findings provide insight into the relationship between interaction structure and thermodynamic behavior and suggest that appropriately chosen construction paths may serve as a basis for efficient extrapolation strategies toward the thermodynamic limit.