Abstract
A higherorder topological insulator (HOTI) in two dimensions is an insulator without metallic edge states but with robust zerodimensional topological boundary modes localized at its corners. Yet, these corner modes do not carry a clear signature of their topology as they lack the anomalous nature of helical or chiral boundary states. Here, we demonstrate using immunity tests that the corner modes found in the breathing kagome lattice represent a prime example of a mistaken identity. Contrary to previous theoretical and experimental claims, we show that these corner modes are inherently fragile: the kagome lattice does not realize a higherorder topological insulator. We support this finding by introducing a criterion based on a corner chargemode correspondence for the presence of topological midgap corner modes in nfold rotational symmetric chiral insulators that explicitly precludes the existence of a HOTI protected by a threefold rotational symmetry.
Introduction
Topology and geometry find many applications in contemporary physics, ranging from anomalies in gauge theories to string theory. In condensed matter physics, topology is used to classify defects in nematic crystals, characterize magnetic skyrmions, and predict the presence or absence of (anomalous) metallic states at the boundaries of insulators and superconductors^{1,2}. For the latter, the topological nature of the boundary modes, be they pointlike zero modes^{3,4,5}, onedimensional chiral^{6,7} and helical states^{8,9,10,11,12}, or twodimensional surface Dirac cones^{13,14,15,16,17,18}, resides in their robustness. One can only get rid of these states by a bulk bandgap closing and reopening or by breaking the protecting symmetry, which can be either an internal or a crystalline symmetry. For example, in twodimensional topological insulators^{8,9,10,11,12} one can gap out the helical edge states by introducing a Zeeman term that explicitly breaks the protecting timereversal symmetry. Similarly, one can move the end states of a SuSchriefferHeeger^{19} chain away from zero energy by breaking the chiral (sublattice) symmetry at the edges and/or in the bulk.
Recent theories exploiting the protecting role of crystalline symmetries have led to the discovery of the socalled higherorder topological insulators^{20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36} (HOTI): states of matter characterized by the presence of topologically protected modes living at the D − n dimensional boundary of a D dimensional insulator, with n > 1 denoting the order. Thus, a twodimensional secondorder topological insulators features pointlike corner modes, while a threedimensional secondorder topological insulator features helical or chiral edge modes along the onedimensional edges. The prediction of higherorder topological insulators has triggered an enormous interest in scanning material structures and engineering metamaterials, e.g. electric circuits^{28}, exhibiting topological corner or hinge modes. However, in identifying higherorder topological insulating phases different complications arise. First, as in conventional firstorder topological insulators, the system can display ordinary ingap boundary states^{37} that are not the prime physical consequence of a nontrivial bulk topology. Second, D − 2 boundary modes of a Ddimensional system can be a manifestation of the crystalline topology of the D − 1 edge rather than of the bulk: the corresponding insulating phases have been recently dubbed as boundaryobstructed topological phases^{38} and do not represent genuine (higherorder) topological phases. These complications are particularly severe in secondorder higherorder topological insulators in two dimensions, since the corresponding zerodimensional topological boundary modes fail to possess the anomalous nature chracteristic of, for instance, onedimensional chiral modes or zerodimensional Majorana corner modes in secondorder topological superconductors. Singling out a proper secondorder topological insulator in two dimensions thus represents a task of exceptional difficulty. In this paper, we prove this assertion by showing that one of the first model system suggested to be a secondorder topological insulator in twodimension^{39,40,41,42,43}—the breathing kagome lattice model—does not host any higherorder topological phase. The corner modes experimentally found in this system^{40,41,43} are neither engendered from a secondorder bulk topology nor from the edge topology characterizing boundaryobstructed topological phases. They instead are an example of conventional corner modes, with the exact same nature of the edge modes generally appearing in onedimensional insulating chains^{44}. We contrast the fragility of the corner modes in the kagome lattice with the robustness of the corner modes in chiralsymmetric insulators, which possess the full immunity of the topological corner modes of a HOTI when an additional evenfold rotational symmetry is present. We also formulate a onetoone correspondence between fractional corner charges^{35,45} and corner modes, which predicts the presence or absence of topological zero modes in chiral symmetric insulators with a \({{\mathcal{C}}}_{n}\)rotational symmetry, and explicitly precludes the appearance of a HOTI phase protected by a threefold rotational symmetry.
That boundary ingap modes can be encountered in both a topological and a nontopological host system is immediately shown by analazying the simple example of onedimensional band insulators, which we recall here to simplify the discussion of our results below. Let us consider specifically a paradigmatic minimal model for a onedimensional band insulator: the RiceMele model^{46}. It is schematically shown in Fig. 1a. The electrons living on the red—sublattice A (green—sublattice B) sites experience an onsite energy +m (−m), can hop within the unitcell with hopping amplitude t, and between adjacent unitcells with a hopping amplitude \({t}^{\prime}\). Within the bulk energy gap and when \( t/t^{\prime} \, <\,1\) we find that the left edge hosts a state at energy +m, whereas the right edge hosts a state at energy −m [c.f. Fig. 1b]. On the contrary, when \( t/t^{\prime} \, >\,1\) the system fails to exhibit any boundary state [c.f. Fig. 1c]. Hence, the RiceMele model exhibits boundary states in half of the parameter space, assuming the termination shown in Fig. 1a is chosen. In order to establish whether these boundary states are topological in nature, we resort to the main characteristic of topological boundary states: their robustness against smooth perturbations. We notice the following:

(i)
The energies of the ingap boundary states change either introducing smooth deformations of the bulk Hamiltonian or applying edgespecific perturbations. An example of this is illustrated in Fig. 1b.

(ii)
The boundary states can completely dissolve into the bulk bands without an accompanying bulkband gap closing, as shown in Fig. 1c. In the Rice–Mele model this occurs for \( t/t^{\prime}  =1\).

(iii)
A tailormade edge potential can even lead to the creation of boundary states out of the bulk bands in the parameter region \( t/t^{\prime} \, >\,1\), as explicitly shown in Fig. 1d.
The boundary states of the RiceMele model fail to exhibit any kind of robustness, and can be therefore qualified as ordinary, i.e., nontopological boundary states.
This, however, is not yet the end of the story. In fact, in onedimensional insulating models it is possible to exploit the effect of the internal particlehole and chiral symmetries. Their existence implies that the model Hamiltonian anticommutes with an (anti)unitary operator that squares to 1 (±1). The Bogoliubovde Gennes (BdG) Hamiltonian describing a superconductor is, by its very definition, particlehole symmetric. However, both the particlehole and the chiral symmetry also play a role outside the superconducting realm. They can in fact arise as approximate symmetries of the effective model Hamiltonian describing an insulator. As an example, in the RiceMele chain we may set m = 0, in which case the Hamiltonian anticommutes with σ_{z}. The resulting model is the wellknown SuSchriefferHeeger (SSH) chain.
Using the results above, we find that the SSH chain displays a left and a right boundary state at zeroenergy if and only if \( t/t^{\prime} \, <\,1\). Importantly, these boundary states now represent truly topological boundary states. Obviously, it would amount to cherrypicking if these states would be dubbed as topological only because they are at zero energy. The rationale for the above is instead based on the following fact: The boundary modes cannot be moved away from zero energy using chiral symmetry preserving bulk or edge perturbations, as long as these do not close the bulk band gap. For example, longrange hoppings between the two sublattices do not move the boundary states since the chiral symmetry is preserved when these processes are included. Hence, the boundary states of the SSH chain are robust zeroenergy modes that are protected by the internal chiral symmetry.
Moreover, and as shown in the Methods section, we can topologically discriminate between systems with an odd and even number of left (right) edge states pinned at zero energy. Therefore, onedimensional chiralsymmetric insulators—but the argument above holds true also for particlehole symmetric insulators belonging to the class D of the AltlandZirnbauer table^{47,48}—are characterized by the group \({{\mathbb{Z}}}_{2}=\{0,1\}\), with the identity element 0 corresponding to chains featuring an even number of left edge states, and the element 1 corresponding to chains featuring an odd number of left edge states. Physically, the \({{\mathbb{Z}}}_{2}\) group law translates into the fact that if one combines (i.e. hybridizes in a symmetrypreserving way) a chain with an even number of left edge states and a chain with an odd number of left edge states, the combined twoleg atomic chain possesses an odd number of left edge states. On the contrary, hybridizing two chains with both an even or odd number of left edge states results in a twoleg atomic chain with an even number of left edge states. In Fig. 2 we illustrate this by considering a twoleg atomic chain consisting of two SSH chains, each of which featuring a single left edge state, hybridized in two different ways. In the first example, shown in Fig. 2a we have hybridized one SSH chain that terminates with an A site, with another SSH chain that terminates with a B site. The result of this hybridization, is that the two left edge states also hybridize, and move away from zero energy, as explicitly shown in Fig. 2c. This is simply a manifestation of level repulsion. However, the twoleg atomic chain shown in Fig. 2b displays a more interesting behavior. There we have hybridized two SSH chains that both terminate with the same sublattice. Despite the hybridization, we find that the two edge states do not move away from zero energy [c.f. Fig. 2d]. Even though these two examples provide purely anecdotal evidence, they do suggest that the physics of zeroenergy edge states is not fully captured by a \({{\mathbb{Z}}}_{2}\)invariant.
This is indeed the case for chiralsymmetric insulators. In these systems, the isolated zero energy modes must be eigenstates of the unitary chiral symmetry, and can be consequently characterized by their chiral charge. For the specific example of the SSH atomic chain, this also implies that an isolated zero mode is either fully localized on the Asublattice or on the Bsublattice. Denoting with \(\left\Psi \right\rangle\) the zero energy end state of a SSH chain and with \(\chi =\left\langle \Psi \right{\sigma }_{z}\left\Psi \right\rangle\) the corresponding chiral charge, we have that if χ = 1 the zeroenergy mode will be fully localized on the sublattice A, whereas if χ = −1 the zero energy mode will be fully localized on the sublattice B. The perfect localization of the end states clearly implies that having at hand two zeroenergy states localized on the same sublattice, and thus with same chiral charge, impedes any level repulsion as this would necessarily break the chiral symmetry. On the contrary, a pair of zeroenergy modes localized on different sublattices can be opportunely coupled and moved away from zero energy, in perfect agreement with the features of the twoleg atomic chain shown in Fig. 2c, d. This proves that the physics of zero energy states in chiralsymmetric insulators is encoded in the \({\mathbb{Z}}\)valued chiral charge:
where the sum runs over all left edge states \(\left{\Psi }_{j}\right\rangle\). Note, however, that the relation between the chiral charge χ_{L} and the number of left edge states at zero energy does not represent a onetoone correspondence. With a zero chiral charge, a pair of zero energy boundary states can be still encountered. This thus implies that the absolute value of the left chiral edge charge defines a lower bound for the total number of zeroenergy states, and is equal to the number of edge states modulo 2. Hybridizing two insulators with chiral charges n_{1} and n_{2}, respectively, we will end up with a chain that features at least ∣n_{1} + n_{2}∣ edge states at zero energy.
To summarize, boundary states are a generic feature of onedimensional band insulators. However, in the absence of any symmetry these states are not robust, for instance they can dissolve into the bulk bands, or even created as a result of tailormade edge perturbations. Instead, if one considers systems with a particlehole or chiral symmetry, we find that the parity of the number of left and right edge states is robust against symmetry allowed perturbations. Moreover, for systems with a chiral symmetry we find that there is a chiral charge associated to each edge, which represents a \({\mathbb{Z}}\) number and is a lower bound for the number of edge states.
Results
Corner modes in the breathing kagome lattice
The preceding discussion on the physical properties of end states in onedimensional insulators can be applied in an analogous fashion to corner states in twodimensional insulators. This can be nicely illustrated using the breathing kagome lattice [c.f. Fig. 3a], which can be thought of as the twodimensional cousin of the RiceMele atomic chain. For simplicity, we will assume in the remainder that all intraunit cell hopping amplitudes have an equal magnitude t. We will also make the same assumption for the interunit cell hopping amplitudes (magnitude \({t}^{\prime}\)). In addition to the inter and intraunit cell hopping parameters, we will first allow for different onsite energies in the three sublattices, which we denote with m_{1}, m_{2}, and m_{3}, respectively. We note that in this situation the breathing kagome lattice is in the wallpaper group p1, i.e. there are no crystalline symmetries other than the inplane translations.
Following ref. ^{49} we find that when considering the system in a opendisk geometry, the lower left, lower right, and upper corner host corner states at energies m_{1}, m_{2}, and m_{3} in half the parameter space, i.e., for \( t/t^{\prime} \, <\,1\). This condition being very similar to the one we encountered in the RiceMele atomic chain, suggests that also the corner modes of the breathing kagome lattice correspond to conventional ingap bound states. And indeed these modes dissolve into the bulk by changing the ratio \( t/{t}^{\prime}\), see, e.g., ref. ^{49}. Next, let us consider the situation in which the onsite energies of the three sublattices are constrained to be equal, i.e.m_{1} ≡ m_{2} ≡ m_{3} ≡ 0. In this situation, the lattice is in the wallpaper group p3m1, which is generated by the point group \({{\mathcal{C}}}_{3v}\) with the addition of translations. The fact that the corner modes now reside at zero energy in an extended region of the parameter space could suggest that the corner modes represent in this case genuine topological boundary modes. Contrary to the SSH atomic chain, however, the kagome lattice model does not possess an internal chiral symmetry that can protect the existence of corner modes pinned at zero energy. This follows from the very simple fact that a chiralsymmetric insulator is incompatible with an odd number of sublattices. Nevertheless, recent studies have suggested that other symmetries could protect the existence of the zeroenergy corner modes, qualifying them as topological corner modes and consequently the prime physical consequence of a higherorder nontrivial bulk topology. Specifically, in ref. ^{39} it has been suggested that the corner modes are protected by the combination of the threefold rotation symmetry and the mirror symmetry of the point group \({{\mathcal{C}}}_{3v}\). Instead, in refs ^{40,41,42} a generalized chiral symmetry has been defined in order to prove the topological nature of the corner modes. Such a generalized chiral symmetry is equivalent to requiring that the model Hamiltonian may only be perturbed by hopping processes between different sublattices.
We now show that the opposite is true, and that the corner modes found in the breathing kagome lattice are nothing but conventional boundary modes even for m_{1,2,3} ≡ 0. To simplify our discussion below, we will consider the flatband limit of t = 0. This limit has the main advantage of removing all system size dependence in our analysis. As mentioned in the preceding section, the defining characteristic of any topological boundary mode is its immunity against perturbations that do not close the insulating band gap and preserve the protecting symmetries. In the context of twodimensional insulators (without metallic edge states) in general, and of the breathing kagome lattice in particular, this would mean that the corner modes have to remain pinned at zero energy upon perturbing the twodimensional bulk, the onedimensional edges, or the zerodimensional corners. To see whether the breathing kagome lattice possesses boundary modes with such robustness, we have considered the effect of applying a local perturbation at the three corners. Specifically, we have introduced longrange hopping processes, with amplitudes s_{1} and s_{2}, at the three corners, as schematically shown in Fig. 3b. Note that the corner perturbation fulfills all the symmetry constraints: the threefold rotational symmetry, the mirror symmetries, and the generalized chiral symmetry are all preserved. Let us now consider the evolution of the corner state energy as we adiabatically switch on the perturbation. In Fig. 3c, we have plotted the evolution of the spectrum assuming, as mentioned above, the intracell hopping amplitude t ≡ 0 and the corner perturbation hopping amplitudes satisfy s_{1} ≡ −s_{2}. We immediately find that the corner modes do not remain pinned at zero energy. In fact, we find that upon increasing the perturbation strength the energy of corner mode even crosses the edge and bulk valence bands at energies \({t}^{\prime}\) and \(2{t}^{\prime}\), respectively. This demonstrates that the corner modes in the breathing kagome lattice do not possess any topological robustness. Their presence or absence is not in a onetoone correspondence with a topological invariant. Instead they simply constitute ordinary boundary states, as do the ones occurring when the onsite energies m_{1,2,3} are different from zero.
Even though the local corner perturbation considered above includes longerrange hoppings, we emphasize that the corner modes of the breathing kagome lattice are also unstable against perturbations that involve shortrange processes only. We illustrate this in Fig. 3d, where we consider an edge perturbation that interchanges the values of the intra and intercell hopping amplitudes t and \({t}^{\prime}\) along the lattice boundaries alone. Considering again for simplicity the limit t = 0, we can readily infer that the system will fail to exhibit any ingap corner mode, since the lattice is composed of disconnected dimers whose bonding and antibonding states are at energies \(\pm \!{t}^{\prime}\) and disconnected triangles with energies \(2{t}^{\prime}\) and \({t}^{\prime}\).
We finally wish to point out that the example of the breathing kagome lattice demonstrates that topological corner modes of insulators represent the exception rather than the rule. Therefore, in the absence of any mathematical or physical strong motivation supporting the topological nature of the corner modes, explicit immunity checks should be performed. In particular, these immunity checks should especially involve perturbations that respect all the putative protecting symmetries. In this respect, we note that ref. ^{40} for instance has analyzed only the reaction of the kagome lattice corner modes under perturbations that explicitly broke the protecting symmetries. Furthermore, we emphasize that, as demonstrated above, immunity checks can be simply performed starting out from a flatband, i.e. atomic, limit. Nevertheless, also perturbative approaches can be employed to shine light on the (non)topological nature of the corner modes. Consider for instance, a zero energy corner mode \(\left\chi \right\rangle\) disturbed by a generic perturbation ΔV. For the corner state to be topological, the energy of the corner mode should be pinned to zero to any order in ΔV. Whereas different putative protecting symmetries can enforce the firstorder perturbative correction to vanish, there is a priori no symmetry able to guarantee the absence of a coupling between the corner state \(\left\chi \right\rangle\) and the edge or bulk states \(\left\Psi \right\rangle\). As a result, the second order correction to the corner state energy \({\sum }_{\Psi \ne \chi }\frac{ \left\langle \Psi \right\Delta V\left\chi \right\rangle { }^{2}}{{E}_{\Psi }}\) will generically differ from zero, except when an actual chiral symmetry is present. In fact, the latter guarantees that the finite contribution to the secondorder correction coming from a state Ψ is immediately cancelled out by the contribution due to its chiral partner \(\widetilde{\Psi }\) with energy \({E}_{\widetilde{\Psi }}={E}_{\Psi }\). This further demonstrates the nontopological nature of the corner modes encountered in the breathing kagome lattice.
Corner modes in chiralsymmetric insulators
Having established with a concrete microscopic model that the corner modes in the kagome lattice simply corresponds to fragile ordinary boundary modes, we next introduce a chiralsymmetric insulator featuring robust corner modes. This will also allow us to discuss the different nature and degree of protection provided by a nontrivial edge topology as compared to the bulk topology of a higherorder topological insulator.
The microscopic tightbinding model we will consider is schematically depicted in Fig. 4a. It possesses an internal conventional chiral symmetry and a \({{\mathcal{C}}}_{4}\) fourfold rotational symmetry. When considered in an opendisk geometry that respects both the rotational and the chiral symmetries, the system features four zeroenergy states which are completely localized at the corners of the lattice when the intracell hopping amplitude t ≡ 0. Precisely as the end modes of the SSH chain, these corner modes can be characterized with a \({\mathbb{Z}}\) number corresponding to their chiral charge χ. More importantly, we find that the the corner modes remain pinned at zero energy under the influence of perturbations that preserve the chiral symmetry, and neither close the edge nor the bulk band gap. This follows from the same reasoning which is behind the stability of zero energy modes in the SSH atomic chain. In particular, it should be stressed that this stability does not rely on the fourfold rotational symmetry.
There is, however, an important distinction between the edge states of the SSH chain and the corner states of a twodimensional chiralsymmetric insulator. The presence of edge states in the SSH atomic chain is only dependent on the topology of the onedimensional bulk Hamiltonian. On the contrary, the presence of a corner mode in a twodimensional chiralsymmetric insulator is dependent on both the topology of the neighboring edges and the topology of the twodimensional bulk. To illustrate this point, we apply to the model shown in Fig. 4a an edge perturbation that can be strong enough to close and reopen the edge gap. Specifically, we introduce a perturbation on the top and bottom edges [c.f. Fig. 4b], thereby explicitly breaking the fourfold rotational symmetry. The spectral flow obtained by increasing the strength of the edge perturbation [c.f. Fig. 4d] shows that the zeroenergy modes remain pinned at zero energy in the weak perturbation regime. However, in the strong edge perturbation regime, i.e. after the closing and reopening of the edge band gap, the corner modes disappear. Importantly, the edge perturbation leaves the bulk of the crystal completely intact independent of its strength. This shows that zeroenergy modes carry a halfway protection: they are removable only by edge perturbations causing closing and reopening of the edge band gap. In the language of ref. ^{38}, the chiralsymmetric insulating model shown in Fig. 4a, therefore represents a boundaryobstructed topological phase protected by the chiral symmetry. We note that in these phases it is impossible to disentangle the edge topology from the bulk topology. Therefore, the presence of a zeroenergy corner state by itself does not provide insights in the bulk topology alone.
Even though the chiral symmetry on its own is insufficient to stabilize the corner modes against strong edge perturbations, we find that the additional presence of the fourfold rotational symmetry does offer this kind of protection. To show this, we consider an additional edge perturbation along the left and right edge, increasing the strength of which eventually leads to the configuration shown in Fig. 4c where the \({{\mathcal{C}}}_{4}\) symmetry has been restored. The ensuing spectral flow shown in Fig. 4e shows that an additional edge gap closing and reopening point leads to a revival of the zeroenergy modes. Therefore, the fourfold symmetric system is characterized by zero energy modes independent of the presence of a strong edge perturbation, qualifying them as the manifestation of the twodimensional bulk topology. In other words, supplementing the chiral symmetry with the \({{\mathcal{C}}}_{4}\) symmetry turns the model into a secondorder topological insulator. This can be also seen using the following argument: in the concomitant presence of the chiral and \({{\mathcal{C}}}_{4}\) rotational symmetry, each corner mode is at the intersection of two adjoined edges related to each other by the fourfold rotational symmetry. The chiral charge χ of this corner mode will change to χ → χ + j, with j an integer upon closing an reopening the band gap along one of the two edges. On the other hand, owing to the fourfold rotational symmetry, the same band gap closing and reopening will occur on the other edge, which will thus contribute with an additional and equal change of the chiral charge. Hence, in total we find that the chiral charge of the corner mode is modified as χ → χ + 2 × j. This therefore implies that, by virtue of the \({{\mathcal{C}}}_{4}\) symmetry, the parity of the chiral charge, and consequently the parity of the number of zero modes per corner ν, is invariant under strong edge perturbations and thus represents a proper \({{\mathbb{Z}}}_{2}\)invariant of such a twodimensional insulator. Note that this topology is considerably weaker than the topology of the SSH chain, as the latter is characterized by a \({\mathbb{Z}}\)number and does not necessitate the presence of a rotational symmetry.
Corner chargemode correspondence
We now show that the topological immunity of the zeroenergy boundary modes in chiralsymmetric insulators with a fourfold rotational symmetry can be also proved using a corner chargemode correspondence that can be generalized to chiral insulators with an evenfold rotational symmetry. We first recall that, as shown in refs. ^{35,45}, the crystalline topological indices characterizing rotational symmetric twodimensional insulators are revealed in the fractional part of the corner charge. Specifically, for corners whose boundaries cross at a maximal Wyckoff position with a site symmetry group that contains the nfold rotational symmetry, the fractional part of the corner charge is a topological \({{\mathbb{Z}}}_{n}\) number uniquely determined by the symmetry labels of the occupied Bloch states at the highsymmetry momenta in the Brillouin zone.
Generally speaking, the correspondence between these bulk topological indices and the fractional corner charge is not reflected in the presence or absence of corned modes. However, we now show that a direction relation exist between the fractional part of the corner charge and the parity of zeroenergy corner modes in chiralsymmetric insulators featuring a two, four, or sixfold rotational symmetry. We will in fact demonstrate that the parity of the number of zeroenergy states per corner ν obeys the formula
In the equation above, N_{total} is the total number of sites per unit cell, n, as before, the order of the rotational symmetry, and Q_{v.b.} the corner charge due to the valence bands, which is a quantity quantized to 0 or 1/2 modulo 1. Before deriving the relation above, a few remarks are in order. First, we wish to emphasize that the corner charge Q_{v.b.} should be computed as the total charge within a \({{\mathcal{C}}}_{n}\)symmetric corner region that is congruent with unit cell centers whose site symmetry group contains the \({{\mathcal{C}}}_{n}\) rotational symmetry. This implies that the boundaries of the corner region should be related to each other via the rotational symmetry, and that the boundaries of the region cross at the \({{\mathcal{C}}}_{n}\)symmetric unit cell center. Example of these corners for \({{\mathcal{C}}}_{2}\), \({{\mathcal{C}}}_{4}\), and \({{\mathcal{C}}}_{6}\) symmetric system are shown in gray in Fig. 5a–c. Second, we remark that the direct relation between corner charge and corner modes is strictly valid for geometric configurations in which the finitesize system can be tiled using an integer number of unit cells. This exclude the presence of fractional unitcells in the opendisk geometry. Note that the lattice structures shown in Fig. 5a–c obey this constraint. Finally, we will consider lattices where: (i) the positions in the unit cell with a site symmetry group containing the \({{\mathcal{C}}}_{n}\) symmetry do not host any atomic site and (ii) atomic sites do not intersect the unitcell boundaries. Note that these conditions automatically ensures that the total number of atomic sites per unit cell is a multiple of the rotational symmetry order n.
We can now derive the corner chargemode correspondence of Eq. (2), and first note that at full filling the corner charge is equal to N_{total}/n modulo 2. By inspection of Fig. 5, it can be easily seen that in all three lattice structures N_{total}/n = 1. Next, we note that the corner charge at fullfilling can be decomposed into three separate contributions: a valence band Q_{v.b.}, a conduction band Q_{c.b.}, and finally an ingap state Q_{ingap} contribution. Here, the valence (conduction) band part accounts for all states whose energies E lie below (above) the band gap, i.e. E ≤ Δ/2 (E ≥ Δ/2), with Δ the band gap. The ingap contribution instead is due to states whose energies E lie inside the band gap, i.e., ∣E∣ < Δ/2. Furthermore, the presence of chiral symmetry guarantees that the corner charge due to the conduction band is identically equal to the corner charge due to the valence band, i.e. Q_{v.b.} = Q_{c.b.}. Note that the latter is a true equality of number, which includes also the integer part of the charge. As a result, we obtain the following:
Finally, we use that the ingap contribution Q_{ingap} is equal, modulo 2, to the parity of the number of corner states per corner ν. Hence, upon rearranging terms we arrive at Eq. (2) that proves the correspondence between fractional corner charge and the presence of zeroenergy corner modes. As the corner charge is a direct probe of the twodimensional bulk topology^{35,45}, Eq. (2) implies that the parity of the the number of zeroenergy states per corner ν itself is a manifestation of this bulk topology.
In principle, one may repeat the above analysis for \({{\mathcal{C}}}_{3}\)symmetric insulators that are also chiralsymmetric. However, an interesting interplay between the chiral symmetry and the threefold rotational symmetry unfolds. Namely, the chiral symmetry requires that at halffilling the corner charge is a multiple of 1/2, instead the threefold rotation symmetry implies that the corner charge is a multiple of 1/3. Satisfying both conditions leaves only one option, namely a vanishing corner charge. In other words, the chiral symmetry renders the corner charge trivial in a \({{\mathcal{C}}}_{3}\)symmetric insulator, in the same way that timereversal symmetry renders the Chern number trivial. Therefore, we find that the mere presence or absence of zeroenergy corner states in a \({{\mathcal{C}}}_{3}\)symmetric insulator does not shine any new light on the twodimensional bulk topology. In fact, any \({{\mathcal{C}}}_{3}\)symmetric configuration that can be tiled with an integer number of unitcells will fail to exhibit an odd number of zeroenergy states per corner. To prove this, let us suppose that each of the three corners would host a single zeroenergy state. This would then imply that the lattice as a whole exhibits a chiral imbalance. However, this is at odds with the original assumption that the finite geometry can be tiled with an integer number of unitcells. Ergo, \({{\mathcal{C}}}_{3}\)symmetric insulators fail to exhibit an odd number of zeroenergy modes per corner.
The corner chargemode correspondence shows that in \({{\mathcal{C}}}_{2}\), \({{\mathcal{C}}}_{4}\), and \({{\mathcal{C}}}_{6}\)symmetric insulators the presence of ingap zero modes can be regarded as a particularly simple probe of bulk topology as long as the chiral symmetry is present. Put differently, the presence of ingap zero modes serves as a proxy for the corner charge in chiralsymmetric insulators. In systems where the chiral symmetry is broken such a direct relation does not hold any longer. This, however, does not completely exclude the possibility to probe the bulk crystalline topology using spectral informations because of the presence of the socalled filling anomaly^{45}. The filling anomaly can be understood by noticing that for a \({{\mathcal{C}}}_{n}\)symmetric crystal in an opendisk geometry—as in Fig. 5 we consider finite size crystals that are rotationsymmetric around the unit cell center – we can relate the corner charge to the total number of electrons modulo n via #electrons = n × Q_{v.b.}. On the other hand, an identical crystal with periodic boundary conditions would host N_{F}modulon electrons, as the central unitcell contributes N_{F} electrons, whereas the surrounding ones come in nfold multiples of N_{F}. The filling anomaly arises when the total number of electrons for opendisk geometry does not match with the number of electrons of the corresponding lattice with periodic boundary conditions, i.e.
This also implies that the mismatch between the open and closed lattices is encoded in the difference δ = n × Q_{v.b.} − N_{F}. We next perform a gedankenexperiment where we assume to continuously interpolate between the periodic and opendisk boundary conditions. Specifically, we imagine to start with periodic boundary conditions, and then cut open the twodimensional torus \({{\mathbb{T}}}_{2}\) as shown in Fig. 5d–f. This can be achieved by continuously tuning to zero the amplitudes for hoppings crossing the black cutting lines in Fig. 5d–f. During this process δ states will cross the Fermi level E_{F} either from below or above (depending on the sign of δ). Indeed, this is the only way to resolve the filling anomaly. For the \({{\mathcal{C}}}_{4}\)symmetric model studied above, we find that upon tuning the system from a torus to an opendisk precisely two modes cross the Fermi level, see Fig. 5(g). Note that in the absence of a chiral symmetry there is nothing that prevents the ingap states crossing E_{F} from dissolving into the bulk valence or conduction bands at the end point of the spectral folow. In other words, to probe the crystalline topology of a rotationsymmetric crystal via spectral methods one should track the evolution of the spectrum throughout the entire opening of the torus \({{\mathbb{T}}}_{2}\). While the described cutting procedure may seem rather complex, we envision that it can be implemented for artificially created lattices. We would like to emphasize that the cutting procedure can also be used away from halffilling. Hence, the spectral flow is able to convey more information regarding the crystalline topology than the detection of an ingap zero mode does. Finally, we note that also this spectral tool cannot be used to probe the crystalline topology of \({{\mathcal{C}}}_{3}\)symmetric lattices for the very simple reason that a triangular geometry cannot be obtained from opening a torus. Thus, for \({{\mathcal{C}}}_{3}\)symmetric insulators there is no spectral signature of the bulk crystalline topology, and one can only resort to corner charge probes.
Discussion
To sum up, we have shown that precisely as boundary modes in atomic chains, corner modes are a generic feature of twodimensional band insulators and are not necessarily a signature of topology. Contrary to previous theoretical and experimental claims, we have indeed proved that the corner modes encountered the breathing kagome lattice do not exhibit any kind of topological robustness, and have to be instead qualified as ordinary boundary modes. We have also contrasted the fragility of these modes with the robustness of the zeroenergy modes appearing in insulators equipped with an internal chiral symmetry. When taken alone, the chiral symmetry provides the halfway topological robustness characteristic of the recently introduced boundaryobstructed topological phases. Furthermore, the presence of a rotational symmetry provides an additional protection mechanism that qualifies the zero energy modes as the prime physical consequence of a higherorder bulk topology.
We have also proved that the immunity of the topological corner modes in the concomitant presence of rotational and chiral symmetry directly follows from a onetoone correspondence between fractional corner charges, which reveal the crystalline topology of generic insulators, and the parity of the number of zero modes. This onetoone correspondence only works in crystals possessing an evenfold rotational symmetry, and thus exclude \({{\mathcal{C}}}_{3}\)symmetric crystals such as the kagome lattice. We wish to remark, however, that even though the breathing kagome lattice does not display a genuine bulkcorner correspondences, its underlying crystalline topology is still reflected in the fractional charge at corners or other topological defects, such as dislocations^{50}.
In closing, we would like to highlight that the topological immunity of zero energy modes in insulators is very different in nature from the one provided by timereversal symmetry in conventional" firstorder topological insulators. This is immediately apparent from the fact that the helical edge states of a twodimensional topological insulator represent anomalous states: Their anomaly reside in the fact that it is impossible to find a onedimensional insulator with an odd number of Kramer pairs at the Fermi energy. On the contrary, the zeroenergy fermionic modes encountered in chiralsymmetric insulators do not represent an essential anomaly for the very simple reason that a quantum dot with an odd number of fermionic modes is entirely allowed in nature. This difference is not only interesting per se. In fact, a topological boundary mode that is also anomalous carries an additional degree of protection. Consider for instance a twodimensional topological insulator whose edges are brought in close proximity to a timereversal invariant nanowire. In such combined system the edge boundary modes will survive, at odds with what would happen in a SuSchriefferHeeger atomic chain if the termination is changed by an atomic site addition. Moreover, one can also proceed in the opposite direction. For example, one may modify a twodimensional insulator that is both \({{\mathcal{C}}}_{3}\)symmetric and chiralsymmetric but fails to exhibit any ingap mode by attaching single sites to its three corners. Even though this would result in the presence of robust zeroenergy corner modes, it should be stressed that these modes are nothing but a microscopic detail of the corner and edges. Put differently, these modes are not in any way informative of the bulk topology.
Methods
Topological zero modes in chiral symmetric atomic chains
The robustness of the zero modes in a chiralsymmetric atomic chain can be rigorously proved using the following: let H(λ) parametrize the chiralsymmetric perturbed Hamiltonian (with λ the perturbation strength) of a semiinfinite atomic chain that features a single left end state \(\left{\Psi }_{L}(0)\right\rangle\) at zero energy. To show that this edge state will remain pinned at zero energy, we can track the left edge state \(\left{\Psi }_{L}(\lambda )\right\rangle\), assuming λ is switched on adiabatically. The chiral symmetry guarantees that every state \(\left\chi \right\rangle\) at an energy +E ≠ 0 has a chiral partner \({\sigma }_{z}\left\chi \right\rangle\) at energy −E. Hence, end states can only move away from zero energy in pairs. Thus, we can conclude that the parity of the number of left edge states pinned at zero energy is robust against all continuous perturbations that respect the chiral symmetry. Note that although this conclusion is strictly valid for a semiinfinite chain, it also holds for finite chains, i.e., with a left and a right edge, as long as the decay length of the edge states is small compared to the system size. In particular, this argument cannot be used when the perturbation closes the bulk band gap, since in the latter case the decay length of the boundary states always reaches the system’s size. This additionally shows that the presence of the zeroenergy edge states is an intrinsic feature of the onedimensional bulk Hamiltonian.
For completeness, we finally note that, being a bulk quantity, the \({\mathbb{Z}}\) chiral charge χ_{L} of the topological zero modes can be expressed as a winding number in terms of the onedimensional Bloch Hamiltonian
using the following formula:
For a detailed derivation of the above bulkboundary correspondence, we refer the reader to ref. ^{51}.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
C.O. acknowledges support from a VIDI grant (Project 68047543) financed by the Netherlands Organization for Scientific Research (NWO).
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van Miert, G., Ortix, C. On the topological immunity of corner states in twodimensional crystalline insulators. npj Quantum Mater. 5, 63 (2020). https://doi.org/10.1038/s41535020002657
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