Phase transitions in the Lieb-kagome lattice interpolation.

We study the ferromagnetic Ising model on a two-dimensional transition lattice that continuously interpolates between the Lieb and kagome lattices via a geometric deformation parameterized by the angle θ. The exchange couplings follow a distance-dependent decay controlled by an exponent n, enabling us to disentangle the interplay between geometry and interaction range. Using extensive Monte Carlo simulations and Binder cumulant analysis, we map the phase diagram T_{c}(θ) and show that the critical temperature increases monotonically as the lattice acquires higher effective connectivity. In the large-n limit, the system approaches a nearest-neighbor regime in which the Monte Carlo critical temperatures converge to the known exact values of the Lieb and kagome limits. Finite-size scaling and data collapse confirm that the entire interpolation belongs to the two-dimensional Ising universality class. Beyond demonstrating the robustness of the critical behavior across the Lieb-kagome transition, the present framework suggests natural extensions. The continuous geometric interpolation between the nonfrustrated Lieb lattice and the frustrated kagome lattice offers a suitable platform to explore antiferromagnetic interactions, where the interplay between frustration and long-range couplings may induce nontrivial ordering.