Corrections to Kubo Formulation of Elastic Stiffness

Employing the Kubo linear response formalism to calculate the elasticity of anisotropic systems has been shown to yield odd elastic moduli. For Hamiltonian systems, this result seems to be contradictory as it would violate energy conservation. To resolve this discrepancy, we examine the results of quantum linear response in the context of our expectation from classical elasticity theory. We demonstrate that the geometry of the space of strain perturbations introduces correction factors to the correspondence between the Kubo formula and the elastic moduli which resolves the contradiction. We demonstrate this concretely using a model for a quasi-2D quantum Hall system in a tilted field. We discuss the implications of our results for interpreting more general linear response functions.