High-temperature modulations, mesoscale interactions, and hyperscaling breakdown in Ising models with frustration: Some insights from thermodynamic geometry

In this work we revisit the axial third-nearest-neighbor Ising chain and examine in detail some aspects of its phase behavior ensuing from competing interactions and resulting frustration. We probe the phase behavior with two complimentary tools: a microscopic two-point correlation function C(n), which we carefully construct after appropriate spin transformations, and a macroscopic, thermodynamic curvature R which we obtain from the free entropy. We report observations of phenomena such as hyperscaling breakdown and intermediate-ranged modulations among others. The zero-field thermodynamic curvature R_{0} is shown to systematically subdivide the ground-state phases into regions of attractive or repulsive effective interactions of varying strength. Furthermore, R_{0} brings forth the significance of some third-order moments in describing the effects of frustration, including the multiphase lines. Combined use of both the tools in the short-ranged modulated order regime confirms and further clarifies the discussion in [S. Khatua et al., Phys. Rev. E 111, L022105 (2025)10.1103/PhysRevE.111.L022105] regarding the appropriate measure of high temperature correlation length in this regime.