Fig. 4: Hierarchical construction of octupole and hexadecapole insulators using synthetic dimensions.
From: Higher-order topological insulators in synthetic dimensions
a Unit cell and b tight-binding model of the octupole insulator. Thin lines have coupling strength γ, and thick lines have coupling strength λ. Blue and red lines represent positive and negative coupling strengths, respectively. c Dipole insulator (SSH strip) formed using two polarization modes in a single resonator25. EOM1 introduces a frequency offset between the resonances of the two polarizations. EOM2 is modulated by a signal similar to Eq. (1) to form the synthetic frequency dimension spanned by ωm. d Unit cell of the quadrupole insulator formed by coupling two site rings (green) with an auxiliary link ring (gray) with a slightly smaller length and asymmetrically placed with respect to the coupling region with an offset η+ = π/2β041. This implements layer 1 of the octupole unit cell in a. The modulation phases between adjacent site rings differ by π. e Implementation of layer 2 of the octupole unit cell in a. The red link ring implements a negative coupling in real space by having an offset η− = 3π/2β0. Negative-valued real-space coupling strengths are needed in our construction of octupole and hexadecapole insulators. f 2D lattice of modulated rings with a synthetic frequency dimension forming the 3D octupole insulator in b. g Array of rings implementing the unit cell of the hexadecapole insulator using two layers of the octupole insulator vertically coupled. The phases of all synthetic and real-space couplings alternate between the two layers. The vertical ring couplings are positive-valued. h 4D hypercubic unit cell of the hexadecapole insulator. The inner cube is realized using the bottom layer in g, and the outer cube is realized using the top layer
