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We investigate relations between spatial properties of the free energy andthe radius of Gaussian curvature of the underlying curved lattice geometries.For this purpose we derive recurrence relations for the analysis of the freeenergy normalized per lattice site of various multistate spin models in thethermal equilibrium on distinct non-Euclidean surface lattices of the infinitesizes. Whereas the free energy is calculated numerically by means of the CornerTransfer Matrix Renormalization Group algorithm, the radius of curvature has ananalytic expression. Two tasks are considered in this work. First, we searchfor such a lattice geometry, which minimizes the free energy per site. Weconjecture that the only Euclidean flat geometry results in the minimal freeenergy per site regardless of the spin model. Second, the relations among thefree energy, the radius of curvature, and the phase transition temperatures areanalyzed. We found out that both the free energy and the phase transitiontemperature inherit the structure of the lattice geometry and asymptoticallyapproach the profile of the Gaussian radius of curvature. This achievementopens new perspectives in the AdS-CFT correspondence theories.

Free Energy Analysis of Spin Models on Hyperbolic Lattice Geometries