Mixed response geometry and critical crossover in the Ising model

Abstract

We develop a geometric formulation of thermodynamic response in interacting spin systems and apply it to the two-dimensional Ising model. Treating inverse temperature and magnetic field as coordinates on a thermodynamic control manifold, we show that the mixed response field $\hat{I}{\text{c◯}}_{{\hat{I}}^{2}h}=-N\mathrm{cov}(m,e)$ arises naturally as a curvature-like quantity that measures correlations between magnetic and energetic fluctuations. Monte Carlo simulations reveal a strongly localized mixed-response ridge that emerges from the critical point and extends into the finite-field crossover regime. Analysis of the susceptibility, specific heat, and mixed-response maxima demonstrates distinct scaling behavior in the magnetic, energetic, and mixed fluctuation sectors. When represented in normalized response coordinates, trajectories obtained at different magnetic fields collapse onto a common curve, indicating that the evolution of the mixed response is strongly constrained by the susceptibility. This collapse suggests the emergence of a low-dimensional response manifold and points toward a geometric description of critical crossover based on relations among response functions rather than equilibrium states alone. The framework establishes a direct connection between fluctuation correlations, critical scaling, and geometric thermodynamic response.