Extended Data Fig. 6: Fractional mode density in large trivial and deformed systems.

a, b, Tight-binding simulation of fractional mode density in large topologically trivial system (15 unit cells per side) with a Frank angle Ω = −90°. We set the intercell coupling strength to be 0.2 and intracell coupling strength to be 1. a, Simulated DOS spectrum and real-space distribution of in-gap modes. The in-gap modes are localized within the central defective unit cell, which only has three sites. b, Fractional portion of the mode density for each band. The mode density takes an integer value in all unit cells. c–e, Tight-binding simulation of fractional mode density in a large system (15 unit cells per side) with a Frank angle Ω = −90° and with the central unit cell trivialized. We set the intercell coupling strength to be 1 and intracell coupling strength to be 0.2. c, Schematic of deformed lattice with trivialized central unit cell. d, Simulated DOS spectrum and real-space distribution of in-gap modes. On-site potentials are applied to the corner sites as well as the ‘interior corners’ to pull six topological bound states into the bulk bandgap. e, Fractional portion of the mode density for each band. The colour of each unit cell indicates the fractional portion of mode density in that cell, and the area of each dot is proportional to the mode density of the corresponding resonator. The fractional part of the integrated mode density around the central unit cell remains 3/4 for bands 1 and 3 and 1/2 for band 2.