Self-Assembly of Isostatic Self-Dual Colloidal Crystals

Self-dual structures whose dual counterparts are themselves possess unique hidden symmetry, beyond the description of classical spatial symmetry groups. Here we propose a strategy based on nematic monolayer attractive half-cylindrical colloids to self-assemble these exotic structures. This system can be seen as 2D semidisks. By using Monte Carlo simulations, discover two isostatic self-dual crystals, i.e., an unreported crystal with $pmg$ space-group and twisted kagome crystal. For approaching critical point, find double degeneracy full phononic spectrum at point merging tilted Weyl nodes into one critically Dirac node. The latter is ``accidentally'' located high-symmetry line. formation this unconventional node due emergence flatbands which linear combinations ``finite-frequency'' floppy modes. These modes understood mechanically coupled rhombus chains vibrating in some uncoupled ways. Our work paves way for designing fabricating materials mechanical or properties.