BBH3 Model: Asymmetric Chiral Corner States

• BBH3 model is a two-dimensional higher-order topological lattice characterized by a six-sublattice tight-binding Hamiltonian with π-flux that yields chiral corner states.
• Its redefinition of sublattices uncovers unconventional chiral corner states that persist as topological bound states in the continuum even when embedded in the bulk spectrum.
• The model employs a novel topological invariant via normalized edge polarization, establishing an edge-corner correspondence that overcomes limitations of conventional bulk-corner links.

Searching arXiv for the specified paper and closely related BBH / higher-order topology context. [arXiv search unavailable in this interface; proceeding from the provided paper metadata and details.] The BBH3 model is a two-dimensional higher-order topological model introduced to study asymmetric systems with chiral boundary states. It is constructed by stacking SSH3 chains and their mirror-symmetric counterparts and by introducing a $\pi$-flux in each plaquette. In the formulation reported in "Nontrivial topology in one- and two-dimensional asymmetric systems with chiral boundary states" (Li et al., 25 Sep 2025), the model supports chiral corner states that are not protected by spatial symmetries such as mirror or inversion, but instead become manifest after an appropriate redefinition of sublattices. In asymmetric regimes, these corner states can persist even when their energies are embedded in the bulk spectrum, exhibiting the characteristics of topological bound states in the continuum (TBICs) (Li et al., 25 Sep 2025).

1. Lattice construction and Hamiltonian

The BBH3 model is defined as a six-sublattice tight-binding system. Its general Bloch Hamiltonian is given by (Li et al., 25 Sep 2025)

$$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$

with

$h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$

where the $t_i$ are hopping amplitudes.

The symmetric and asymmetric limits are determined directly by these couplings. When $t_1=t_2$ and $t_3=t_4$, the model has full mirror symmetry; otherwise it is asymmetric. This construction places the BBH3 model in the broader class of higher-order lattice systems while distinguishing it from conventional quadrupole models through its explicit accommodation of asymmetry.

A central motivation for the model is that conventional topological classifications are strongly organized by symmetry, whereas the topology of asymmetric systems, especially in two dimensions, is comparatively less developed. The BBH3 model therefore serves as a concrete setting in which chiral boundary topology can be identified without relying on the usual spatial-symmetry protection.

2. Chiral corner states and their spectral character

A principal result is the existence of new chiral corner states in the BBH3 model (Li et al., 25 Sep 2025). These states are localized at system corners and are described as fundamentally distinct from conventional quadrupole-protected corner modes. Their defining feature is that they are tied to a chiral structure revealed in an appropriately redefined basis rather than to mirror or inversion symmetry in the original lattice description.

In the mirror-symmetric BBH3 model, for parameters crossing a topological transition, fourfold degenerate chiral corner states appear in the bandgap, with real-space support sharply localized to certain corners. Upon breaking mirror symmetry, these states split into two sets of twofold degenerate states, each appearing at different energies and at different critical parameter values. The reported behavior is notable because these states can emerge without bulk gap closing.

The asymmetric regime yields a further distinction: chiral corner states can remain localized even when their energies lie inside the bulk bands. In that case they exhibit the characteristics of topological bound states in the continuum. This directly counters the common assumption that corner localization in higher-order systems must be associated with isolated in-gap eigenvalues. Here, localization and topological character survive even when spectral embedding occurs.

The wavefunction structure is also highly constrained. Each chiral corner state is strictly zero on a subset of the redefined sublattices, which is presented as a hallmark of sub-symmetry protection. This exact amplitude suppression is one of the clearest operational signatures distinguishing these states from more conventional higher-order corner modes.

3. Redefinition of sublattices and sub-symmetry

The method used to uncover the chiral nature of the BBH3 corner states is based on redefining sublattices (Li et al., 25 Sep 2025). The procedure begins by diagonalizing subspaces associated with part of the unit cell, for example by taking a subset of four out of six sublattices and finding a representation in which these degrees of freedom are decoupled. This produces a new basis consisting of linear combinations of the original orbitals, denoted schematically by states such as $|1'\rangle$ and $|2'\rangle$.

Under this transformed basis, the Hamiltonian decomposes into coupled copies of 2D Rice-Mele models or their generalizations. Within that representation, the origin of the corner states becomes transparent: a chiral corner state is supported only on a single redefined sublattice, while all amplitudes on the remaining redefined sublattices vanish exactly. The explicit condition quoted for a state $|C_\alpha\rangle$ is

$$\langle \beta | C_\alpha \rangle = \delta_{\alpha\beta}.$$

This basis dependence is not presented as a mere change of notation, but as the mechanism by which a hidden protecting structure becomes visible. The relevant protection is termed sub-symmetry, and the resulting states are described as sub-symmetry-protected corner states. This suggests that the topological content of the model is not exhausted by the manifest symmetry of the original lattice graph; rather, it depends on an effective decomposition that reorganizes the Hilbert space into topologically meaningful sectors.

The same conceptual framework is stated to generalize ideas from the one-dimensional SSH3 setting to more complex two-dimensional geometries. A plausible implication is that distinct asymmetric lattices may share a common topological origin once expressed in an appropriate redefined-sublattice basis.

4. Topological invariant $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$0 and its two-dimensional extension

The BBH3 analysis employs a new topological invariant, denoted $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$1, introduced first in one dimension and then generalized to two dimensions (Li et al., 25 Sep 2025). In one dimension, the invariant is the normalized Zak phase constructed from a normalized reduced wavefunction derived from a Rice-Mele-like effective Hamiltonian:

$$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$2

where $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$3 is mirror-symmetrized via normalization.

For the BBH3 model in two dimensions, the relevant quantity is the normalized edge polarization, defined for an edge ribbon with momentum $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$4 and normalized edge state $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$5 by

$$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$6

The index $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$7 labels the edge band.

The invariant is reported to be quantized, typically as $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$8 or $$\mathcal{H}_{\mathrm{BBH3}}(k_x,k_y)= \begin{bmatrix} 0 & t_1 & h_1 & h_2 & 0 & 0 \ t_1 & 0 & t_2 & 0 & 0 & 0 \ h_1^* & t_2 & 0 & 0 & 0 & -h_2 \ h_2^* & 0 & 0 & 0 & t_2 & h_1 \ 0 & 0 & 0 & t_2 & 0 & t_1 \ 0 & 0 & -h_2^* & h_1^* & t_1 & 0 \end{bmatrix},$$9, whenever the underlying effective Hamiltonian after normalization possesses the appropriate mirror symmetry. Abrupt changes in $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$0, such as a transition from $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$1 to $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$2, correspond exactly to the appearance or disappearance of chiral edge or corner states. Crucially, this correspondence is stated to hold even in the absence of bulk or edge gap closing.

The role of $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$3 in the BBH3 model is therefore not merely classificatory. It supplies an operational criterion for tracking topological transitions in asymmetric systems where familiar gap-closing diagnostics may fail. The paper further describes the two-dimensional invariant as a $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$4 indicator, in the sense that it detects changes associated with odd additions of corner or edge states.

5. Edge-corner correspondence and the failure of conventional bulk-corner correspondence

For the BBH3 model, the usual higher-order expectation that a two-dimensional bulk invariant should directly predict corner states is qualified. The reported result is that bulk-corner correspondence can fail in BBH3 because of the specific nature of the chiral corner states (Li et al., 25 Sep 2025). Instead, the relevant principle is edge-corner correspondence.

In this formulation, the one-dimensional invariant $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$5 of the edge Hamiltonian predicts the existence of corner modes. The critical points of $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$6 match the appearance of chiral corner states in the open-boundary-condition spectrum. This reframes the corner physics as arising from topological structure inherited from the edges rather than from a direct bulk quadrupole-type index.

This distinction addresses a frequent misconception in the interpretation of higher-order topological phases. In standard symmetry-protected quadrupole systems, corner states are often treated as straightforward descendants of bulk multipole moments. In the BBH3 model, by contrast, the corner states are linked to the topology of effective edge degrees of freedom defined in the restructured basis. Their existence is therefore mediated by a boundary-sensitive invariant rather than by a conventional bulk topological marker.

A plausible implication is that asymmetric higher-order systems may require a hierarchy of reduced or effective invariants, especially when the relevant protected states become visible only after basis redefinition.

6. Acoustic realization and numerical observation

The paper proposes an acoustical realization of the BBH3 model and numerically demonstrates chiral corner states in that platform (Li et al., 25 Sep 2025). In the proposed setup, each sublattice is modeled by an air-filled acoustic cavity of size $h_1=t_4 e^{i k_x}, \qquad h_2=t_3+t_4 e^{-i k_y},$7 cm. Positive and negative hopping terms between cavities are emulated by straight and twisted connecting tubes, which control both coupling strength and sign through the phase of the tube transmission. The unit cell therefore consists of six coupled cavities, mirroring the theoretical structure.

Within this realization, the reported numerical observations are consistent with the lattice model. Bulk bands match the theoretical predictions for the BBH3 model. In finite samples, in-gap corner-localized eigenmodes appear, and their spatial profiles confirm the chiral character by occupying only certain cavities in the unit cell. When asymmetry is introduced by making all tube couplings unequal, the chiral corner states persist even as their frequencies enter the bulk bands, realizing corner TBICs.

The experimental proposal is significant because it connects the abstract redefined-sublattice picture to a physically implementable architecture. The details provided indicate that sign-controlled couplings and multi-site unit cells are sufficient to emulate the required flux pattern and asymmetry. This suggests that the BBH3 model is not confined to formal band-theoretic analysis, but can serve as a concrete platform for probing asymmetric higher-order topology and boundary states embedded in continua.

7. Conceptual significance within asymmetric topological band theory

The BBH3 model exemplifies a broader theoretical claim advanced in the source paper: models with entirely different structures may share the same topological origins once the correct sublattice redefinition is performed (Li et al., 25 Sep 2025). Within that perspective, the BBH3 system is not only a specific higher-order lattice construction, but also a case study in how topology may persist in asymmetric settings where conventional symmetry-based classifications appear insufficient.

Several points summarize that significance. First, the model shows that corner states need not rely on spatial symmetry protection. Second, topological transitions need not coincide with bulk or edge gap closings. Third, corner localization may survive inside the bulk spectrum, producing states with the characteristics of TBICs. Fourth, the effective diagnostic structure shifts from a conventional bulk invariant to an edge-based normalized polarization.

Taken together, these features place the BBH3 model at the intersection of higher-order topology, asymmetric band structures, and hidden-basis formulations of chiral protection. The model’s importance lies less in extending a familiar symmetry-protected template than in demonstrating a different organizing principle: by redefining sublattices, one can expose topological boundary states that are otherwise obscured in the original lattice representation.