Category: Physics

Summary: Mapping the threshold at which defect-localized soft modes appear in disordered elastic materials.

This experiment investigates an open problem in mechanical metamaterials: how much targeted band softening is needed before the lowest vibrational mode in a disordered long-range elastic operator becomes localized around defects rather than spread across the bulk material.

The experiment constructs random symmetric elastic operators of increasing size (64 to 2048 dimensions) with a fraction of sites designated as defects. It then uses bisection search to find the critical softening parameter — the point where the lowest eigenmode transitions from bulk-spread to defect-localized.

By running this across many disorder realizations and system sizes, we build a dense finite-size threshold map that has not been previously published.

Method: Dense eigensolve (np.linalg.eigh) with iterative deepening. System sizes N=64 to 2048, bisection on defect softening parameter, multiple disorder realizations per size.

What is measured: Critical softening threshold, participation ratio, localization transition sharpness across system sizes.