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 { a nematic monolayer of} attractive half-cylindrical colloids to self-assemble these exotic structures. { This system can be seen as a 2D system of semi-disks.} By using Monte Carlo simulations, we discover two isostatic self-dual crystals, i.e., an unreported crystal with pmg {space-group} symmetry and the twisted Kagome crystal. For the pmg crystal approaching the critical point, we find the double degeneracy of the {full} phononic spectrum at the self-dual point, and the merging of two tilted Weyl nodes into one \emph{critically-tilted} Dirac node. The latter is `accidentally' located on the high-symmetry line. The formation of this unconventional Dirac node is due to the emergence of the critical flat bands at the self-dual point, which are linear combinations of \emph{finite-frequency} floppy modes. These modes can be understood as mechanically-coupled self-dual rhomb chains vibrating in some unique uncoupled ways. Our work paves the way for designing and fabricating self-dual materials with exotic mechanical or phononic properties.