Corner states usually borrow protection from symmetry
Robust corner states are zero-dimensional modes that sit at the corners of a two-dimensional structure while the bulk and edges remain extended. In photonics they have been built mainly in second-order topological insulators, including shrunken-expanded lattices and breathing Kagome lattices. In those designs, global crystalline symmetries quantize the relevant bulk or edge indices and constrain where the corner mode can appear.
That protection has a cost. Because the symmetry is global, structural disorder that breaks it can shift the corner eigenfrequency and eventually destroy localization through hybridization with edge or bulk continua. The paper frames this as the motivating contrast: first-order topological boundary states in Chern and Z2 insulators can survive substantial disorder as long as the bulk or mobility gap stays open, whereas symmetry-protected corner states are more fragile. The question the authors pose is whether robust corner localization must depend on global crystalline-symmetry protection at all.
A Dirac-connected flatband is the source of the mode
The proposed mechanism starts from a Dirac semimetal, whose projected band structure contains flatband edge states linking the boundary projections of a pair of bulk Dirac points. The paper describes two routes from this starting point. Lifting the Dirac-point degeneracy opens a topologically nontrivial bandgap in which the Dirac-connected edge states persist and become topological boundary states. The alternative, used here, leaves the degeneracy in place and instead relocates the flatband edge states into a separate, topologically trivial bandgap.
That relocation is achieved by boundary engineering rather than by changing the bulk lattice. In the tight-binding picture, boundary sites carry an additional mass Ms; in the photonic crystal, the same knob is the radius of the outermost row of dielectric cylinders. Shifting the flatband edge state through the upper bulk band and into the trivial gap produces spectrally isolated edge and corner states without invoking a nontrivial bulk invariant.
Shrinking boundary cylinders moves the flatband into the gap
The photonic design is a two-dimensional honeycomb lattice of dielectric cylinders with lattice constant a, radius r = 0.176a, and relative permittivity 11.5, studied for transverse magnetic modes. Surface density-of-states calculations on semi-infinite crystals with open boundaries show the unperturbed flatband edge state lying between the two Dirac degeneracies at K and K', with slight bending from the intrinsic photonic dispersion.
The authors then modify only the outermost row of cylinders. Reducing the boundary radius to rE = 0.8r pushes the flatband edge state upward into the bulk continuum. Reducing it further to rE = 0.50r carries the state across the upper bulk band until it is fully isolated inside the trivial bandgap. This is the central design move: a local geometric change to the boundary, not a global deformation of the lattice, decides whether the edge-derived state sits inside a gap.
Each tailored corner gets its own frequency
Because the shift depends on the local boundary radius, the approach is programmable. In a triangular photonic crystal the authors independently tune the cylinders at the top, left, and right vertices, setting boundary radii of rE = 0.5r, 0.4r, and 0.33r respectively. The computed eigenfrequency spectrum separates bulk modes from corner-localized modes, with normalized eigenfrequencies of 0.413, 0.446, and 0.475 for the left, top, and right corners.
Field distributions at those frequencies confirm strong spatial confinement at the corresponding corners. The paper notes that edge states of the finite lattice are not involved, since only the corner units are modified while bulk and edges stay unchanged. The result is deterministic control over both the number of corner modes and their spectral positions inside the bandgap, achieved without altering global lattice geometry or bulk topology.
Microwave measurements confirm a corner mode at 7.48 GHz
The experimental sample is a triangular photonic crystal with lattice constant a = 17 mm and 11 units along each side, built from nonmagnetized yttrium iron garnet cylinders with relative permittivity 11.5, relative permeability 1, and radius r = 3 mm. The three corner cylinders are reduced to r1 = 1.4 mm. The structure is sandwiched between flat aluminum plates acting as effective perfect electric conductor boundaries, excited by a broadband quasi-plane wave from a horn antenna, and probed with a monopole antenna connected to a vector network analyzer over 5 to 12 GHz at 0.02 GHz resolution.
The calculated bulk band structure shows a trivial bandgap between the second and third bands. Measured transmission at the bulk center reveals a bandgap from 6.89 to 8.83 GHz, consistent with the simulations. At the top corner, a prominent transmission peak appears at 7.48 GHz, and near-field scanning at that frequency confirms strong field confinement at the top corner. Bulk-mode field distributions measured at 6.5 GHz and 9.2 GHz provide the comparison. The authors attribute minor simulated-measured frequency discrepancies to fabrication tolerances in the cylinder dimensions.
Disorder stability is the claim that separates this design
The paper's distinguishing claim is that these corner modes remain stable under both position and radius disorder, whereas symmetry-protected corner states undergo rapid bandgap closure and lose corner localization under comparable disorder. This is the comparison that matters for readers weighing the approach against established second-order topological designs: the relevant axis is not whether a topological invariant exists, but whether the localized mode survives perturbations that break a protecting symmetry.
Two scope conditions travel with the experimental result. The two bottom corners were not efficiently excited under plane-wave illumination, because the bandgap contains no bulk or edge transport channels that could deliver energy to them; the measurement therefore confirms one corner mode rather than all three. The authors also report extending the same local geometry strategy to a topological Kagome lattice, with tailored corner states constructed numerically and observed experimentally, so the mechanism is not confined to the trivial-gap honeycomb case.
Compare disorder tolerance before topological pedigree
For all-dielectric light localization, the paper's practical priority is a design trade: it gives up global symmetry protection and nontrivial bulk topology in exchange for corner modes that the authors report as disorder-resilient and spectrally programmable through a single local parameter, the corner cylinder radius. The demonstrated tuning range runs from rE = 0.5r to 0.33r across three corners, producing distinct normalized eigenfrequencies of 0.413, 0.446, and 0.475, and the measured trivial gap spans 6.89 to 8.83 GHz with the corner mode at 7.48 GHz.
The judgment a reader should carry away is conditional. If the requirement is a corner mode whose frequency can be set by local geometry and whose localization is claimed to survive position and radius disorder, this mechanism is the relevant comparison against shrunken-expanded and breathing Kagome designs. If the requirement is that every corner of a finite structure be excited and verified under plane-wave illumination, the reported experiment covers only the top corner, and the bottom-corner limitation follows from the absence of in-gap transport channels rather than from the corner design itself.
Sources used in this answer
Observation of robust corner states in photonic crystals without global symmetries
Locally shrinking corner cylinders in a topologically trivial honeycomb photonic crystal relocates Dirac-connected flatband edge states into a trivial bandgap, producing corner-localized modes that the authors report remain stable under position and radius disorder, with a measured corner transmission peak at 7.48 GHz inside a 6.89-8.83 GHz gap.