Topological signatures in the curvature-induced energy response

Relativistic effective field theory predicts a topological energy response to a gravitational field that appears at third order in spatial gradients. Here, we investigate how this response emerges in the nonrelativistic Haldane model using a microscopic lattice formulation of curvature-induced deformations. We find that the leading first-order energy response is nonuniversal and depends on the bond-resolved structure of the deformation; in particular, it vanishes for a symmetric modulation of the three nearest-neighbor hoppings. In contrast, the third-order response exhibits a discontinuity across the topological transition whose magnitude agrees with the relativistic gravitational Chern--Simons prediction. Thus, although the absolute response is nonuniversal, its third-order discontinuity is universal and retains a characteristic topological fingerprint beyond the relativistic limit.