Being global crystallographic defects, local operations cannot remove disclinations [55]. One may use the Volterra method [56] to construct a disclination. An example is depicted in Figure 1A, where a sample is cut into a few identical wedge portions, and one (marked in yellow) is removed to form a disclination after gluing the remaining sections without lattice mismatch. According to the homotopy theory, a disclination is characterized by two parameters (Ω, B). Here Ω is the Frank angle, whose magnitude is the wedge angle and whose sign indicates adding or removing a wedge, and B is the Burgers vector, which measures the lattice distortion induced by the defect [57, 58]. Choosing a start point, B can be evaluated by comparing the loop path around the disclination core and the loop path in a defect-free sample. For more details of the calculation of the Buregers vector, please refer to the Supplementary Material. For a square lattice respecting C ₄ point group symmetry, Ω can only be a multiple of π/2. The group of non-equivalent classes of B is isomorphic to the discrete group Z ₂ and Z ₂ ⊗ Z ₂ for Ω = ±π/2 and ± π, respectively [42]. The details of equivalenece classes of B is discussed in the Supplementary Material.

To concrete our study, we consider the 2D SSH model [59, 60], one of the typical models that admit topological corner states [26, 27, 60–62]. A sample of the 2D SSH model is depicted in Figure 1B, where the unit cell consists of four sub-lattices forming a square Bravais lattice. There are two types of hopping, namely, the intra-cell hopping γ and the inter-cell hopping γ′. Depending on the ratio of |γ/γ′|, the 2D SSH model can be in the atomic insulator phase or the atomic-obstructed phase. For the atomic insulator phase, its Wannier center coincides with the atomic lattice, for the atomic-obstructed phase, its Wannier center locates at the middle of two unit-cells. It is noted that for the atomic-obstructed phase, the Wannier center cannot be changed untill the band gaps close. For the detials of the band structure and fractional charge of the 2D SSH model, please refer to the Supplementary Material. For |γ| < |γ′| as in Figure 1B, the lowest energy band is inverted at (π/a, 0) and (0, π/a) in the reciprocal-space (with a the lattice constant) and becomes topologically nontrivial accompanying with corner states [59]. The appearance of topological corner states in the 2D SSH model is owing to the shift of dimerized cells as displayed by the light magenta square in Figure 1B, whose centers are related to the vectored Zak’s phase p = (p _x , p _y ) by a factor of a 2 π [63–66]. Constrained by the periodicity of Bravais lattice, p _x/y is defined within 0,2 π and becomes a quantization of π when inversion symmetry is present, as determined by the parity of the bulk wave function at (0,0) and (π/a, 0)/(0, π/a) in the reciprocal space. Upon shifting the center of dimerized cells as well as Wannier states, the lowest energy band accommodates less than one electron in the unit cells located at the edges and corners, known as the filling anomaly that results in topological edge and corner states carrying 1/2 and 1/4 fractional charges, respectively [7].

Figure 1C displays two distinct disclinations with Ω = −π/2 for the 2D SSH model, where the square represents the unit cell, and the intra-cell and inter-cell hoppings are omitted. Depending on the unfolded Burgers vector B in undistorted space (indicated by red vectors in Figure 1C), the disclinations of Ω = −π/2 are classified into two topologically distinct types as labeled by s = (0, 0) and s = (1/2, 1/2), respectively. The relation between B and s is given as s = 1 2 ( 2 B ) mod 2 , which forms a bijection to the homotopy group of B and thus is a real-space topological invariant. For a finite sample with full point-group symmetry, s can also be determined by counting the number of unit cells along the boundaries of the sample, i.e., s = 1 2 [ ( Γ x , Γ y ) mod 2 ] , where Γ _x and Γ _y denote the numbers of unit cells on x- and y-boundaries, respectively. It is noted that s forms a one-to-one mapping to nonequivalent disclination centers. For well-localized bound states without resonance, we focus on the cases that |γ − γ′| > min (|γ|, |γ′|), i.e., γ, γ′ = 1.0, 3.0 and γ, γ′ = 3.0, 1.0, where band gaps form between the first and the second bands, and the third and the fourth bands.

Considering that the removal or addition of the wedge part resolves the filling anomaly at the disclination center, we expect a concurrent action of the real-space topological invariant s and the reciprocal topological invariant p, which we propose as an additive rule between them. In Table 1, s is tabulated for all possible values of Ω for the 2D SSH model. The integers inside Table 1 are the numbers of bound states at the different types of disclination centers for both trivial and nontrivial reciprocal topologies. From Table 1, we see that even for the trivial reciprocal topology, bound states exist as s + p/2π is nontrivial, whereas for the nontrivial p bound state is missing if s + p/2π is trivial. We define the net topology of real-space and reciprocal topology as P = ( s + p / 2 π ) mod 1 , and discuss three unique manifestations of the proposed additive rule in the follows, which embody the content in Table 1. Extending the additive rule to other lattice models is possible, and we discuss it in the latter part.

Previous studies suggest that the relationship between real and reciprocal spaces should be multiplicative [21, 36, 67, 68]. We obtain the additive rule because we focus on the bound states rather than the fractional charge. As discussed in Ref. [36], the fractional charge at the disclination core is given by the formula Q = Ω 2 π ( n b + 2 n c ) + T ⋅ p , where Q is the fractional charge at the disclination center, n _b , n _c are the numbers of the inverted band at high symmetric k points, and T = a ₁ d ₁ + a ₂ d ₂ with d _i ⋅e _j = δ _ij . As suggested by the formula, Q always appears as finite no matter the real-space topology if nontrivial p exists, which is considered the bulk-disclination correspondence. As demonstrated below, the bound state can appear at the disclination core even for trivial p and disappear for nontrivial p. In other words, the disclination-bound states and trapped fractional charge are dissociated.

The first phenomenon of the proposed additive rule is the dissociation of fractional charges from bound states. The construction of disclination lattices and the caculation of fractional charge is discussed in the Methods section. We consider the samples with − π 2 -disclinations. Figures 2A–C show the fractional charges and bound states for the − π 2 -disclinations with three distinct additive conditions between the real and reciprocal topological invariants s and p. In the left panels of Figures 2A–C, each unit-cell’s numerical datum of charge distribution are written as digits. The bound states are indicated by the dark magenta shades (circles and triangles), and the fractional charges with ±1/4 are marked with the cyan crescents. In the middle panels of Figures 2A–C, we have also displayed the numerical datum of eigenfunctions when electrons are mostly localized for the corresponding left samples at disclination centers. In the right panels of Figures 2A–C, the eigenenergies distributions for samples of left panels are displayed.

As can be seen in the left panel of Figure 2A, fractional charges appear at the disclination center and the sample corners, but bound states are absent at the center (see also the right panel of Figure 2A) even with the nontrivial reciprocal topology p. This result can be intuitively understood using the dimerization of sites as shown by lighter magenta squares in the left panel of Figure 2A. As explained earlier, the corner state accompanying with 1/4 fractional charge appears due to dimerized cells shifting from the original Bravais lattice and the resulting filling anomaly. However, here in Figure 2A, the filling anomaly at the disclination center that is supposed to be induced by nontrivial p is canceled out by the nontrivial real-space topological invariant s. As a result, no fractionally filled dimerized cell is isolated from the bulk states, as suggested by the additive rule between s and p.

Figure 2B shows the disclination with trivial s = (0, 0) but non-trivial p = (π, π). Since the additive rule gives nontrivial P , both the bound states and fractional charges simultaneously appear at the disclination center together with the corner state, as seen in Figure 2B. The eigenenergies distributions are gapless in Figures 2A, B owing to the nontrivial p, where edge states appear within the band gaps. Figure 2C shows a complementary example, where the real-space topology is nontrivial, and the reciprocal space topology is trivial. The additive rule gives nontrivial P . Thus, the bound state appears at the center of disclination without corner states, as shown in Figure 2C. A pseudo fractional charge is just located at the disclination center. We shall note that this fractional charge at the disclination center is further smeared out beyond the fractionally filled dimerized cell as seen in the left panel of Figure 2C, unlike those in Figures 2A, B. It is also noted that in the right panel of Figure 2C, the disclination-bound state appears in the middle of the first energy band gap.

As the emergence of disclination-bound states is due to the dimerization at the disclination core, it is worth discussing the robustness of these bound states. Here we consider two types of perturbations. One is the perturbation without respecting the C ₄ point group symmetry, and another is the perturbation respecting the C ₄ point group symmetry. For the first type of perturbation, we consider three possibilities: onsite potential on the disclination center sites, a dangling bond in the disclination center, and inter-cell hopping connecting sites belonging to the same sub-lattice. As detailed in the Supplementary Material, for the perturbations without C ₄ point group symmetry, the amplitude of perturbations cannot go beyond |γ − γ′|; otherwise, the disclination-bound states disappear. For the second type of perturbations respecting C ₄ point group symmetry, the amplitude of perturbations can go beyond |γ − γ′|. This is because of the unique real-space structure in the disclination core, where one sublattice is missing in the central dimer of disclinations that disclination-bound states cannot mix with bulk states respecting C ₄ point group symmetry. It is noted that the disclination-bound states are not located at zero energy, which suggests the absence of chiral symmetry in the formation of disclination-bound states [45, 69, 70].

The second phenomenon of the proposed additive rule is the formation of half-bound states, which decay on one side of the sample but extend over the other. Here we consider a disclination structure with a unsymmetric s index, i.e., (s _x , s _y ) = (0, 1/2) for Ω = −π as displayed in Figure 3. A bound state can be viewed as a wave function with a purely imaginary wavenumber for all independent real-space directions. Because of the unsymmetric disclination structure between k _x and k _y directions, a half-bound state can be expected. As displayed in Figures 3A, B, we find such half-bound states in our numerical calculations. Interestingly, the decaying direction for the half-bound states depends on the summation value of s + p/2π. As displayed in Figure 3A, when s _y + p _y /2π is nontrivial, the half-bound state decays along the x side. While s _x + p _x /2π is nontrivial, the half-bound state decays along the y side, as displayed in Figure 3B. This is perhaps because of the spatial distortion induced by the disclination structure.

It is noted that the formation of half-bound states seems analogous to edge states due to the second-order topology. In the 2D SSH model, if the systems have p _x p _y = 0 but p _x + p _y ≠ 0, only edge states exist but no corner state. In the present case, this may be paraphrased: For two-sided systems with s _x s _y = 0 but s _x + s _y ≠ 0, only a half-bound state exists but not a bound state. This half-bound state can potentially control wave propagation using artificial crystalline structures such as photonic crystals. These states are impervious to the system size as shown in the Supplemntal Material. For the practical realization of the half-bound state, the hopping amplitude should depend on the distance between the two sites. In this case, the lattice distortion induced by the disclination cannot be ignored. The site’s position should be carefully tuned to achieve a situation similar to the tight-binding model.

The third phenomenon of the proposed additive rule is the hybrid-bound state, which can be numerically observed in any disclination with Ω ≥ π and s _x ≠ s _y . Figure 4A shows a disclination with Ω = π and s = (1/2, 0). This disclination is formed by inserting two extra π/2 blocks into the sample. Considering there are only two-independent directions in two dimensions, we can regard there are three x-parts and three y-parts arranged alternately in Figure 4A. For the sample of Figure 4A, we only observe bound states rather than half-bound states. This is because there are multiple x-parts, unlike the case in Figure 3, which only has one x-part. Furthermore, P is nontrivial regardless of p being trivial or nontrivial. We call this type of disclination-bound state hybrid-bound states because of their unsymmetrical s index. Figure 4B displays the energy spectrum for the disclination in Figure 4A with p = (0, 0), where a doubly degenerate bound state emerges within the band gap. Interestingly, for p = (π, π), the bound state is robust to the onsite potential perturbation as shown in the Supplementary Material, which may be useful for constructing cavities. A full spectrum of parameter pumping for such a hybrid-bound state is also given in Supplementary Material.