Bulk-local-density-of-state correspondence in topological insulators

In the quest to connect bulk topological quantum numbers to measurable parameters in real materials, current established approaches often necessitate specific conditions, limiting their applicability. Here we propose and demonstrate an approach to link the non-trivial hierarchical bulk topology to the multidimensional partition of local density of states (LDOS), denoted as the bulk-LDOS correspondence. In finite-size topologically nontrivial photonic crystals, we observe the LDOS partitioned into three distinct regions: a two-dimensional interior bulk area, a one-dimensional edge region, and zero-dimensional corner sites. Contrarily, topologically trivial cases exhibit uniform LDOS distribution across the entire two-dimensional bulk area. Our findings provide a general framework for distinguishing topological insulators and uncovering novel aspects of topological directional band-gap materials, even in the absence of in-gap states.

The photonic band structure and the topological properties of the isotropic photonic crystal can be well approximated by the tight-binding model (see Fig. S1a), whose Hamiltonian is as follows Here h 12 = t a + t b exp(ik x ), h 13 = t a + t b exp(−ik y ), h 21 = t a + t b exp(−ik x ), h 24 = t a + t b exp(−ik y ), h 31 = t a + t b exp(ik y ), h 34 = t a + t b exp(ik x ), h 42 = t a + t b exp(ik y ), h 43 = t a + t b exp(−ik x ).k = (k x , k y ) is the Bloch momentum.The tight-binding parameters t a , t b reflect the intra-and inter-cell couplings between the neighboring rods, respectively.Here, we neglect the higher-order couplings such as the next-nearest-neighbor coupling.This approximation is valid except for the lowest band which will not change the topology of the band structure as shown in Fig. S1b-c.If the intercell coupling is larger than the intracell coupling, the tight-binding model is in topologically nontrivial insulating phase and there will be topological edge states and corner states under the open boundary condition [1].We can also add perturbations on the onsite energy of edge lattice sites to 'drag' the ingap topological edge states into the bulk spectrum (see Fig. S1c).Here we choose t a = 1.05, t b = 1.6.When onsite energy perturbation E Edge = 1 is added for the edge lattice of the finite chain [2], the edge states are now pulled from the middle of the directional bandgap into the bulk bands.Nevertheless we can observe a multidimensional partition of LDOS for topologically nontrivial phase which clearly demonstrate the bulk-LDOS correspondence for the tight-binding model.
To calculate the LDOS, we first obtain the eigenfunctions of real space lattices and corresponding eigenfrequencies by diagonalizing the Hamiltonian of the finite-size array.We can add up those wavefunctions by their eigenfrequencies to obtain the real space distribution of LDOS.We define the local density of states of the bulk, edge and corner region as depicted in Fig. S1d-i.According to the Figs.S1d-f, at the lower range of the eigenfrequencies, the eigenstates are nearly uniformly distributed at the whole lattice sites for any frequencies for the trival case.However, for the nontrivial lattice as shown in Figs.S1g-i, the LDOS of eigenstates is firstly distributed mainly at the bulk lattice sites, avoiding the edge and corner lattice sites, when the energy of states is low.As we increase the energy, the LDOS of eigenstates is mainly distributed at both the bulk and edge sites, avoiding the corner sites, representing the coexistence of bulk and edge states at this energy.Furthermore, as we continuously increase the energy of states, we find that the LDOS is now mainly distributed at both bulk and corner sites, avoiding edge sites which indicating the coexistence of both bulk and corner states at this energy.The above character in tight-binding models clearly reveals the multidimensional partitions of LDOS for topologically non-trivial insulating phases which consists with the case in photonic crystals.
Next, we add random onsite energy perturbations to the whole lattices.As shown in Fig. S2, Even the disorder breaks the C 4 symmetry of the lattice and the LDOS is not symmetrized, we can also observe a similar multidimensional partition of LDOS for topologically nontrivial phase and a single-dimensional partition of LDOS for topologically trivial phase.Therefore the bulk-LDOS correspondence can distinguishing different topological phases in topological insulators characterized by the displacement of Wannier centers.
Besides topological boundary states, defective edge localized states may also emerge by adding a large onsite energy potential to the boundary lattice sites in the two-dimensional systems.However, these kind of boundary localized defective states originate from the detune of boundary sites to the bulk sites resulting in several fundamental differences from the topological boundary state which are also reflected in their LDOSs.First, topological boundary states stem from re-hybridizations of multiple bands while the defective edge states come from the single band.As a consequence, as shown in Figs.s6a  and s6c, the number of topological edge states and corner states are 2 and 1 respectively at each boundary (the total number in a finite square structure is 8 and 4 due to the four-fold rotational symmetry).However, when adding a large onsite energy potential to the boundary sites of a trivial lattice, as shown in Figs.s7a and s7b, the number of emergent edge and corner localized states are 4 and 3 respectively at different frequencies in the s-band for a specific M and therefore can be distinguished by looking at the spectrum of the LDOS.Secondly, The LDOSs for topological boundary state and defective boundary localized states are also different in real space.As shown in Figs.s6c (Fig. s7c), the LDOS for topological edge (corner) states is strongly localized at the edge (corner) sites and the sites belong to the same sublattice in the next-nearest edge (corner) cells.However, for defective edge (corner) localized states, the LDOSs are extended to other sites in the edge (corner) cells and do not confined in one sublattice.These difference between topological boundary states and defective boundary localized states in the LDOS at both energy domain and real space enable us to identify distinct topological phase by looking at the LDOS characters which is just like the cases with complete bandgaps [2].We also verify these differences in photonic crystals which is consist with the tight binding models.

Fig. S1
Fig. S1 Multidimensional partition of LDOS for topologically nontrivial tightbinding model.a The finite-size lattice with perturbed edge sites with directional bandgap and no in-gap states.b The bulk band structure with directional bandgap.c The projected band structure with no in-gap edge states.d-fThe LDOS of topological lattice at energy E = −5.9084,E = −2.9688,and E = 0.76803 respectively.g-i The LDOS of trivial lattice at energy E = −5.9127,E = −1.779,and E = 0.53095 respectively.

Fig. S2
Fig. S2 Multidimensional partition of LDOS for topologically nontrivial tightbinding model with random disorders.a The added random onsite perturbation at each site.b-c The finite size lattice with random disorders for topologically trivial and nontrivial lattice.d-f The LDOS of topologically trivial lattice at energy E = −5.9517,E = −3.458,and E = −0.20782respectively.g-i The LDOS of topologically non-trivial lattice at energy E = −5.9358,E = −2.925,and E = −0.071543respectively.

Fig. S3
Fig. S3 Photograph of the sample.The boundary of PCs and the air gap is indicated as the dashed line, ∆ = 0.25a is the thickness of the air gap.The lower right panel is a zoom-in picture of the unit cell where d denotes the distances of the dielectric cylinders from the center of the unit cell both horizontally and vertically.The picture shown here is the sample of d = 0.17a with the onsite potential disorder.

Fig
Fig. S4 Schematic of sampled points.Unit cells and the dielectric cylinders are represented as gray squares and darker circles.The blue and pink regions denote the edges and corners of the sample, respectively.Each unit cell is divided into 10 × 10 equivalent areas with orange dots locate at the center of each area which are the measured points.

Fig. S5
Fig.S5The difference between the LDOSs and the excited field distributions.a The simulated LDOS of an eigenstate at f = 3.3045GHz.b-d The simulated excited field distributions at f = 3.3045GHz with a point source placed at the center of the side, the center of the bulk and the corner of the finite-size structure respectively.e The simulated LDOS of a eigenstate at f = 5.5239GHz.f-h The simulated excited field distributions at f = 5.5239GHz with a point source placed at the center of the side, the center of the bulk and the corner of the finite-size structure respectively.i The simulated LDOS of a eigenstate at f = 5.9534GHz.j-l The simulated excited field distributions at f = 5.9534GHz with a point source placed at the center of the side, the center of the bulk and the corner of the finite-size structure respectively.

Fig. S6
Fig. S6 The difference between the defective edge states and topological edge states.a The projective band structures of the tight-binding models for topological edge states (represented by number 1) with intracell coupling ω = 1 and intercell coupling v = 2, and the defective edge states (represented by number 2 and 3) induced by adding large onsite potential (M edge = 1) on the edge sites of trivial lattice (ω = 2, and v = 1).b The local density of states (LDOS) for topological edge states (represented by number 1) and defective edge states (represented by number 2 and 3).c Numerical calculated projective band structures of the photonic crystals for the topological edge states (represented by number 1) and the defective edge states (represented by number 2 and 3) which are similar to the tight-binding models.d LDOS for photonic topological edge states (represented by number 1) and defective edge states (represented by number 2 and 3).

Fig. S7
Fig. S7 The difference between the defective corner states and topological corner states a a The evolution of eigenmodes of the topological corner states (represented by number 1) and b the defective edge states (represented by number 2 and 3, 4) in a trivial lattice under the change of different onsite energy potential (Mcorner) on the four corner sites.c LDOS for topological corner states is strongly localized at the corner sites (marked by 1); LDOSs for defective corner localized states are extended to other sites near the corner sites at the corner unit cells (marked by 2, 3, and 4).d Calculated LDOSs for similar case as those in (c) in photonic crystals.