Bound states at disclinations: an additive rule of real and reciprocal space topology

Focusing on the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model, we propose an additive rule between the real-space topological invariant s of disclinations (related to the Burgers vector B) and the reciprocal-space topological invariant p of bulk wave functions (the vectored Zak phase). The disclination-induced bound states in the 2D SSH model appear only if (s + p/2π) is nonzero modulo the lattice constant. These disclination-bound states are robust against perturbations respecting C 4 point group symmetry and other perturbations within an amplitude determined by p. Besides the disclination-bound states, the proposed additive rule also suggests that a half-bound state extends over only half of a sample and a hybrid-bound state, which always have a nonvanishing component of s + p/2π.

Osaka University, Toyonaka 560-8531, Japan Topological materials are renowned for their ability to harbor states localized at their peripheries, such as surfaces, edges, and corners.Accompanying these states, fractional charges appear on peripheral unit cells.Recently, topologically bound states and fractional charges at disclinations of crystalline defects have been theoretically predicted.This so-called bulk-disclination correspondence has been experimentally confirmed in artificial crystalline structures, such as microwave-circuit arrays and photonic crystals.Here, we demonstrate an additive rule between the real-space topological invariant s (related to the Burgers vector B) and the reciprocal-space topological invariant p (vectored Zak's phase of bulk wave functions).The bound states and fractional charges concur at a disclination center only if s + p/2π is topologically nontrivial; otherwise, no bound state forms even if fractional charges are trapped.Besides the dissociation of fractional charges from bound states, the additive rule also dictates the existence of half-bound states extending over only half of a sample and ultra-stable bound states protected by both real-space and reciprocalspace topologies.Our results add another dimension to the ongoing study of topological matter and may germinate interesting applications.
Unlike edge states, topological corner states usually appear as bound states in the continuum of bulk spectrum, which complicates their experimental detection [27][28][29][30].However, higher-order topological phases induce fractional charges and bound states at a disclination center of crystallographic defects, which has been experimentally observed recently [31][32][33][34].The correlation between the fractional charges carried by bound states appearing at disclination centers and the reciprocal topological invariant of bulk is framed as the bulk-disclination correspondence [35][36][37][38][39].
Inspired by such observations, here we focus on the correlation between these anomalous bound states and the real-space topology of disclinations (in contrast to the reciprocal-space topology of bulk).We find that the real-space topology of disclinations and the reciprocal-space topology of bulk are additive, in the sense that bound states form at a disclination only if the sum between them is nontrivial.This additive rule can be manifested in various ways, amongst which are the dissociation of the appearance of fractional charges from the formation of bound states, the existence of half-bound states extending over only half of a sample, and the transition from bound states protected by real-space topology to those by reciprocal-space topology, which we detail in what follows.Our results bridge the classical real-space topologies of crystalline defects to the reciprocal-space topologies of quantum wave functions and may well fertilize interesting physical phenomena and applications.
Being global crystallographic defects, disclinations cannot be removed by local operations [40].
To construct a disclination, one may employ the Volterra method [41].An example is depicted in Fig. 1a, whereby 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.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 [42,43].For square lattice that respects C 4 point group symmetry, Ω can only be a multiple of π/2 and 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 [37].
To exemplify the aforementioned additive rule between the real-space topology of disclination and the reciprocal-space topology of the wave function of bulk, we consider the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model [44], which is one of the typical models that admit topological corner states [22,23,[45][46][47].A sample of of the 2D SSH model is depicted in Fig. 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 |, it settles in one of two topologically distinct phases.For |γ| < |γ | as in Fig. 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 [44,47].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 Fig. 1b, whose centers are related to the vectored Zak's phase p = (p x , p y ) by a factor of a 2π [48][49][50][51].Constrained by the periodicity of Bravais lattice, p x/y is defined within [0, 2π) 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 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.[6,47].
, 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 xand yboundaries, respectively.
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 I, s is tabulated for all possible values of Ω for the 2D SSH model [52].The integers inside TABLE I are the numbers of bound states at the different types of disclination centers for both trivial and nontrivial reciprocal topologies.From TABLE I, 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 topologies as P = 2(s + p/2π) mod 2, and discuss three unique manifestations of the proposed additive rule in the follows, which embody the content in TABLE I.
The first manifestation is dissociation of fractional charges from bound states.We demonstrate this phenomenon for 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 Figs.2a-c, the numerical datum of charge distribution for each unit cell are written.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 right panels of Figs.2a-c, we have also displayed the numerical datum of eigenfunctions when electrons are mostly localized for the corresponding left samples at disclination centers.
As can be seen in the left panel of Fig. 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 Fig. 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 Fig. 2a.As explained earlier, the corner state accompanying 1/4 fractional charge appears due to dimerized cells shifting from the original Bravais lattice and the resulting filling anomaly.However, here in Fig. 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, there is no fractionally filled dimerized cell isolated from the bulk states as indicated by the additive rule.We note that the fractional charge appears at the disclination center in Fig. 2a simply because of the missing of a site in the central unit cell of the sample.
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 corner states as seen in Fig. 2b. 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, both the bound state and 1/4 fractional charge appear at the center of disclination without corner states as can be seen in Fig. 2c.
We shall note that the fractional charge at the disclination center is further smeared out beyond the fractionally filled dimerized cell as seen in the left and right panels of Fig. 2c.
The second manifestation is the formation of half-bound states, which decay on one side of the sample but extend over the other.The half-bound state exists if s x differs from s y , i.e., with asymmetry between the x and y-directions.According to TABLE I, we need to consider disclinations with Ω = −π, and (s x , s y ) = (0, 1/2) or (1/2, 0).The disclination of this class is shown in Figs.3a and b, where Ω = −π and s = (0, 1/2), but for different p (γ/γ = 1/3 in a and γ/γ = 3 in b).In this case, net topology P is nontrivial for both p = (0, 0) and p = (π, π).
Thus the half-bound states appear as shown in Figs.3a and b, where electrons decay in one direction but extend to the other direction.The decaying direction of half-bound state depends on which component of P is nontrivial, i.e., nontrivial P y yields a half-bound state decaying over the x-side.The formation of half-bound states is analogous to the formation of 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 [45].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 of the system size [47].
The third manifestation can be 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, which has three x-sides and three y-sides arranged alternatingly.Owing to the additional blocks of π/2, the wave function may decay in multiple directions (even when only the x-side is nontrivial), and bound state forms rather than the half-bound state.Furthermore, P is nontrivial regardless of p being trivial or nontrivial.Hence, at the disclination center, bound states form invariably for arbitrary parameters.Figure 4b displays the energy spectrum for the disclination in Fig. 4a with p = (0, 0), where doubly degenerate bound state emerges within the band gap.Interestingly, for p = (π, π), the bound state becomes a singlet with a symmetric wave function, as illustrated in Fig. 4c.This phenomenon reflects on the different topological origins of the bound states.The doublet bound state originates from the real-space topology, which distinguishes the three non-equivalent π/2 blocks of y-side in the real space.On the other hand, the singlet bound state does not differentiate those blocks in real space owing to its reciprocal topological origin.The difference between doublet and singlet bound states reminds us the conjugation relation between real and reciprocal spaces.The bound states at such disclinations are ultra-stable and protected by both real-space and reciprocal-space topologies.They are useful for nano-scale photonic cavities.A full spectrum of parameter pumping for such the bound states is given in Supplement [47].
Finally, we remark on the number of bound states at a disclination as listed in TABLE I for p = (0, 0).For Ω = −π and s = (1/2, 1/2), there are two bound states with opposite energies due to chiral symmetry.For Ω = ±π/2 and s = (1/2, 1/2), there is a pair of degenerate bound states due to the extra nontrivial π/2 block.For Ω = π, there are two pairs of degenerate states with opposite energies due to chiral symmetry and the additional blocks.
To summarize, we have demonstrated an additive rule between the real-space topology and the reciprocal-space topology at typical crystallographic defects, namely the disclinations.The real-space topology is characterized by the parity of the Burgers vector while the reciprocal-space topology by the vectored Zak's phase.We demonstrate three unique phenomena due to the additive rule: the dissociation of the bound state formation and the appearance of fractional charges, the existence of half-bound states, and ultra-stable bound states protected by real-and reciprocalspace topologies.Our results shed further insight into crystalline topology and may usher in novel applications.

This work is supported by the Research Starting Funding of Ningbo University, NSFC Grant
No. 12074205, and NSFZP Grant No. LQ21A040004.K.W. acknowledges the financial support by JSPS KAKENHI Grant No. JP18H01154, and JST CREST Grant No. JPMJCR19T1. ) and s = (0, 0), respectively.s indicates the rotational center of the sample.
Here we discuss the topological invariant of disclinations.As well known, a disclination is characterized by its Frank angle Ω and Burgers vector B. For given Ω, B can be evaluated by a holonomy along a closed path surrounding the disclination center.An example of B is displayed in Fig. S1 (a).Note that B depends on the starting (ending) point of the path and is hence not unique to the disclination.Nevertheless, it is easy to see that the parity of 2B, which we denote by 2s, is independent of the path and indeed unique.It univocally indicates the center of the wedge used in the Volterra construction of a disclination.For a sample possessing full point-group symmetry prior to Volterra process, s locates the rotational center of the sample and 2s can be obtained by counting the number of unit cells (mod 2) along each axis.One may see that, for Ω = ± π 2 , s ∈ {(0, 0), (1/2, 1/2)}.For Ω = ±π, however, s ∈ {(0, 0), (0, 1/2), (1/2, 0), ( 1 Compared with quantum pumping using topological edge states, the parameter pumping here causes spectral flow between bound states protected by real-space topology and those by reciprocal-space topology.As a result, the number of topological transport channels is not limited by the dimensions of systems such as edges and corners. Figure S4: Topological parameter pumping by the disclination with Ω = π and s = (0, 1/2).The green is the energy spectra of the bulk and edge states, and orange is that of the bound states at the disclination.Bound states (including corner states) buried in the spectra of edge and bulk states are not shown.

Figure
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 Burgers vector B (red vectors in Fig. 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 [(2B) 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

FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .Figure
FIG. 1. Construction and characteristic of a disclination.a. Schematic of Volterra process for constructing a disclination.A wedge part spanning angle |Ω| is cut off from a symmetric sample, and the remaining sections are glued without any lattice mismatch.The wedge center is located at the point of rotation symmetry of the sample.The resulting disclination has negative Frank angle Ω = −|Ω|.Alternatively, one may insert an extra wedge instead of removing the wedge, resulting in a disclination with positive Ω = |Ω|.b.Sample of the 2D SSH model in the case of |γ| < |γ | that respects C 4 point group symmetry, where solid/dashed line indicates the intra/inter-cell hopping of strength γ/γ , and square/shade indicates the unit/dimerized cell.c.Two types of disclinations with Ω = −π/2 allowed for samples with C 4 -point group symmetry characterized by s.Each square represents a unit cell, and the lighter ones are the wedges being removed.s is determined by the parity of the numbers of unit cells on the x-and y-boundaries as s = 1 2 [(Γ x , Γ y ) mod 2], which forms a bijection of the homotopy group of the Burgers vector B.

1 D
/2, 1/2)}.Examples are shown in Figure S1 (b) and (c).This observation can also be reached by group-theoretic analysis.Note that B and R θ B − R Ω a + a represent different loops but enclosing the same disclination and should then be deemed equivalent.Here (θ, a) parameterizes a displacement by a and rotation by θ (denoted by R θ ).The equivalence classes of B can be shown to form a Z 2 group for Ω = ±π/2 and a Z 2 ⊗ Z 2 group for Ω = ±π.B.Reciprocal topology of the 2D SSH modelThe Hamiltonian for the bulk 2D SSH model is given asH(k x , k y ) =     0 ρ(k x ) ρ(k y ) 0 ρ * (k x ) 0 0 ρ(k y ) ρ * (k y ) 0 0 ρ(k x ) 0 ρ * (k y ) ρ * (k x ) 0 .Parameter pumping of ultra-stable bound statesIt is worth discussing an adiabatic pumping as a means for exploiting the transition from real-space protected bound states to reciprocal-space protected ones at the disclination.To this end, we set (γ, γ ′ ) = (3 cos θ, 1) with θ being the pumping parameter.As p only depends on the ratio |γ/γ ′ |, we let θ vary from −π/2 to π/2.The pumping spectrum is plotted in Fig.S4.The bound states at the disclination show up as in-gap states.