Entangled four-dimensional multicomponent topological states from photonic crystal defects

Recently, there has been a drive towards the realization of topological phases beyond conventional electronic materials, including phases defined in more than three dimensions. We propose a versatile and experimentally realistic approach of realizing a large variety of multicomponent topological phases in two-dimensional (2D) photonic crystals with quasiperiodically modulated defects. With a length scale introduced by a background resonator lattice, the defects are found to host various effective orbitals of $s$-, $p$-, and $d$-type symmetries, thus providing a monolithic platform for realizing multicomponent topological states without requiring separate internal degrees of freedom in the physical setup. Notably, by coupling the defect modulations diagonally, we report the realization of “entangled” 4D quantum Hall (QH) phases which cannot be factorized into two copies of 2D QH phases, each described by the first Chern number. The structure of this nonfactorizability can be quantified by a classical entanglement entropy inspired by quantum information theory. In another embodiment, we present 4D $p$-orbital nodal lines in a nonsymmorphic photonic lattice, hosting boundary states with an exotic manifold. Our simple and versatile approach holds the promise of topological optoelectronic and photonic applications such as one-way optical fibers.

Figure 1

(a) Dependence of the on-site frequencies of various $s$-, $p$-, and $d$-type defect modes with the defect rod radius $r$. Only $s$-type modes exist for $r<0.2a$, the background rod radius, but progressively more exotic modes appear at larger radii. (b) Illustration of photonic crystal (PC) with background/defect rods colored gray/orange. Slight spatial modulation of the defect radii leads to an emergent entangled 4D QH state described in Fig. 2. (c) The $E_z$ distributions of the various modes in (b). The proliferation of these modes leads to interesting multicomponent effective Hamiltonians.

Figure 2

(a) PBC and (b) OBC frequency dispersion of the entangled defect lattice of Eq. (4), with parameters $r_0=0.1a, r_1=r_2=0.025a, k_w=0$, and rational flux $b=1/4$. Excellent numerical agreement exists between the FEM simulation/TB model [Eq. (5) with $n=4$ along $x$ and $y$; by fitting the band data with Eq. (5), we found that ${\omega }_0=0.36862(c/a), \lambda =-0.014508$, and $t=-0.00398(c/a)]$ results, which are represented by the dotted/continuous curves. With OBC, topological spectral flow exists between the central and top/bottom defect bulk bands. (c) The Berry curvature entanglement spectra ${\tilde{\lambda }}_i$ and entropy for both $xy,zw$ and $xz,yw$ cuts, with long tails indicative of nontrivial entanglement, i.e., nonfactorizability of the second Chern number. An unentangled defect lattice given by $r_{\text{ind}}$ gives a delta-function-like spectrum (red). (d) The $E_z$ distribution of the midgap mode circled green $[k_z=-1.2, \omega =0.3465(c/a)]$ in (b), which is clearly localized at a corner defect rod (thick circle).

Figure 3

(a) Unit cell of our effectively 4D PC with two synthetic dimensions, with red/blue representing positive/negative electric fields. Nonsymmorhic symmetry holds when $k_z=\pm \pi /2$, as indicated by arrows that map the elliptical rods to themselves via the glide reflection $g_y$. (b) Bulk band structure from comsol simulations (blue) and TB model (black) in the $k_z=k_w=\pi /2$ slice, with nonsymmorphic symmetry preserved/broken by tilting half of the rods (top/bottom). Top: A Dirac point degeneracy (purple dot) lies on nodal lines (red) in the 4D BZ, as plotted in the $k_x-k_w$ slice with $k_z=\pi /2, k_y=0$ (inset). Bottom: Degeneracies are all gapped by breaking nonsymmorphic symmetry. (c) Degenerate boundary states in the $k_z=k_w=\pi /2$ slice under $y$-direction OBCs. Left: Spatial profile of boundary degeneracies at $k_x=0$, marked by the red arrow in the comsol band structure (top right), which corroborates with TB results (bottom right). (d) Boundary modes that do not merge into the bulk, shown for slices $k_z=k_w=\pi /2$ (left) and $k_x=0, k_z=\pi /2$ (right). Simulations (top) qualitatively agree with TB results (bottom), with black and red/blue labeling bulk and left/right edge modes, respectively.