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. At third order, the curvature-induced energy response contains a bond-symmetric component together with nonuniversal bond-dependent contributions. The bond-symmetric component, which is isolated by a uniform modulation of the three hoppings, exhibits a universal discontinuity across the topological transition whose magnitude agrees with the relativistic gravitational Chern--Simons prediction. Thus, although the full curvature-induced energy response generally contains nonuniversal lattice contributions, its third-order response retains a universal topological component beyond the relativistic limit.