Disorder perturbation expansion for athermal crystals

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Materials Science

Background:

• Athermal systems lack thermal fluctuations, making their response to disorder unique.
• Understanding particle displacements is crucial for predicting material properties under stress.
• Microscopic disorder significantly impacts the structure and stability of crystalline states.

Purpose of the Study:

• To introduce an exact perturbation expansion for determining displacement fields in disordered athermal systems.
• To analyze the response of crystalline structures to microscopic disorder.
• To provide a hierarchical method for solving force-balance equations to arbitrary accuracy.

Main Methods:

• Developed a perturbation expansion based on the strength of microscopic disorder.
• Introduced a hierarchy of force-balance equations for order-by-order displacement determination.
• Analyzed displacement fields and interparticle forces at linear and second-order approximations.

Main Results:

• Exact displacement fields were determined for an isotropic defect in a crystalline ground state.
• Displacement fields exhibit self-similar properties at each order of the expansion.
• Derived large-distance decay laws: |δr|∼1/r for displacements and |δf|∼1/r² for forces.

Conclusions:

• The perturbation expansion provides an accurate and versatile tool for studying disordered systems.
• The derived displacement fields and force laws offer fundamental insights into defect behavior.
• Results were validated against simulations of soft disk systems, confirming the method's efficacy.