Disorder Perturbation Expansion for Athermal Crystals

We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy minimized configurations of particles interacting through central potentials with microscopic disorder, can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher order solutions. This allows the simultaneous force balance equations to be solved, within a hierarchical perturbation expansion to arbitrary accuracy. We present exact results for an isotropic defect introduced into the crystalline ground state at linear order and second order in our expansion. We show that the displacement fields produced by the defect display interesting self-similar properties at every order. We derive a $|\delta r| \sim 1/r$ and $|\delta f| \sim 1/r^2$ decay for the displacement fields and excess forces at large distances $r$ away from the defect. Finally we derive non-linear corrections introduced by the interactions between defects at second order in our expansion. We verify our exact results with displacement fields obtained from energy minimized configurations of soft disks.