---
title: 'Surface TBICs: Topological Bound States'
url: https://www.emergentmind.com/topics/surface-tbics
type: topic
---

# Surface TBICs: Topological Bound States

Surface topological bound states in the continuum (surface TBICs) are boundary-localized modes whose eigenfrequency or quasienergy lies inside a continuum of extended bulk states, yet which remain spatially confined and non-hybridized. They combine two ingredients that are distinct in ordinary band theory: a topological boundary state, which is guaranteed by a bulk invariant or a symmetry-constrained band inversion pattern, and a BIC mechanism, which suppresses coupling to the available continuum. In the strictest usage, a surface TBIC is a two-dimensional boundary mode of a three-dimensional system. The clearest direct realization reported so far is a three-dimensional phononic crystal supporting a two-dimensional interface-localized TBIC coexisting with a one-dimensional hinge TBIC in the same sample [2509.00344]. Closely related lower-dimensional analogues include edge-localized first-order TBICs in mirror-stacked bilayers [2212.06562] and dispersive edge-localized TBICs in a two-dimensional Floquet-type colored quantum random walk [2503.17263].

## 1. Definition and conceptual boundaries

A BIC is a localized state with its eigenvalue embedded in the continuum of extended states. A TBIC is the special case in which the localized state is topological: its existence derives from a bulk topological phase, while its failure to hybridize with the continuum is enforced by an additional decoupling mechanism or symmetry [2212.06562]. For surface TBICs, the localization occurs on a codimension-1 boundary of the host system, so the canonical case is a 2D surface or interface state embedded in a 3D bulk continuum [2509.00344].

This definition excludes two nearby but distinct cases. First, ordinary topological surface states in projected bulk band gaps are not TBICs, because they are spectrally isolated rather than embedded. Second, conventional surface BICs that rely only on separability or interference are not necessarily topological. The recent TBIC literature makes this distinction explicit: in the three-dimensional phononic-crystal realization, the interface state is topological because it is generated by a valley-Chern-number mismatch, and it is a BIC because it is embedded in the bulk states of an orthogonal subsystem without hybridization [2509.00344].

The term is also used more loosely for lower-dimensional analogues. In the mirror-stacked bilayer construction, the directly demonstrated first-order TBIC is a 0D end or edge state embedded in a 1D bulk continuum, while the higher-order example is a 0D corner state embedded in a 2D bulk continuum [2212.06562]. In the colored quantum-random-walk literature, the reported states are edge-localized dispersive TBICs in a semi-infinite strip geometry rather than literal 3D surface states, but they are still boundary-localized TBICs in the relevant Floquet setting [2503.17263].

## 2. Mechanisms that generate surface TBICs

The most explicit mechanism for genuine surface TBICs is separability. In the three-dimensional phononic-crystal model, the Bloch Hamiltonian can be block-diagonalized as
$$
H_D(\mathbf{k}) = H_2 \oplus H_3,
$$
so the system decomposes into two orthogonal subsystems, denoted \(h^{(2)}\) and \(h^{(3)}\). The \(h^{(3)}\) sector supplies the topological boundary state, while the \(h^{(2)}\) sector supplies the continuum of extended bulk states. Because the eigenstates of the two sectors are orthogonal, a boundary mode of \(h^{(3)}\) can lie inside the bulk spectrum of \(h^{(2)}\) without hybridizing [2509.00344]. In that system, separability is therefore the BIC mechanism, while valley topology provides the boundary state itself.

A second, more abstract mechanism is mirror-sector decoupling. In the mirror-stacked approach, two identical monolayers are coupled through
$$
H=\tau_0\otimes h - t_c \tau_x\otimes I,
$$
with mirror operator
$$
M_z=\tau_1\otimes I.
$$
After a similarity transformation, the Hamiltonian splits into two independent parity sectors,
$$
\bar{H}=h_{\mathrm{even}}\oplus h_{\mathrm{odd}},
\qquad
h_{\mathrm{even}}=h-t_c I,\quad h_{\mathrm{odd}}=h+t_c I.
$$
Each sector inherits the monolayer topology exactly, because the shift by \(\pm t_c I\) changes eigenvalues but not eigenvectors. Tuning \(t_c\) moves the topological boundary state of one sector into the bulk continuum of the opposite sector, while mirror parity forbids hybridization between them [2212.06562]. This construction is not itself a 3D surface-TBIC realization, but it is a universal design principle for converting topological boundary states into TBICs.

A third mechanism appears in discrete-time Floquet-type walks. In the two-dimensional colored quantum random walk, the one-step evolution operator is
$$
\bm U(\theta_1,\theta_2)=\bm T_y \bm R(\theta_2)\bm T_x \bm R(\theta_1),
\qquad
\bm R(\theta)=e^{i\bm S\theta/2},
$$
with a three-dimensional internal “color” space and an SU(3)-generated coin rotation. There is no external periodic drive, but the repeated discrete-time step acts as an intrinsic Floquet evolution. The paper identifies color-induced band mixing among the three internal states as the mechanism underlying the natural formation of Floquet states and the appearance of dispersive TBICs embedded within the bulk of other bands [2503.17263].

These mechanisms illustrate a recurring structural point. In TBICs, the topological protection of the boundary state and the suppression of its coupling to the continuum need not come from the same symmetry. The mirror-stacking work makes this separation explicit: the monolayer topology may rely on inversion or reflection within each layer, whereas the BIC character is enforced by interlayer mirror symmetry [2212.06562].

## 3. Geometries and representative realizations

The strictest surface-TBIC realization currently reported is an airborne three-dimensional phononic crystal composed of a cavity–tube network. Each unit cell contains five cylindrical cavities, and the experimentally realized lattice constants are \(a_0=54\ \text{mm}\) in plane and \(h=27.9\ \text{mm}\) vertically. When two mirrored phases are joined with interface normal to \(y\), the interface lies in the \(XZ\) plane and hosts a two-dimensional interface state. This state is localized near the interface, disperses along both \(k_x\) and \(k_z\), and lies inside the projected bulk continuum of the \(h^{(2)}\) sector. The same sample simultaneously supports a one-dimensional hinge TBIC where the top surfaces of the two phases meet, producing a dimensional hierarchy of 2D surface/interface TBICs and 1D hinge TBICs [2509.00344].

The mirror-stacked bilayer platform provides the most general constructive framework. In the one-dimensional SSH bilayer, the topological end or edge state of one mirror sector can be shifted into the bulk continuum of the opposite sector, yielding a first-order TBIC. In the two-dimensional quadrupole bilayer, the same procedure embeds corner states into the opposite-parity bulk continuum, producing higher-order TBICs. The paper further notes additional BIC phases associated with trivial 1D edge states, including 1D edge states embedded in 2D bulk states, which is especially suggestive for future surface-TBIC constructions [2212.06562].

The Floquet-type colored quantum random walk occupies an intermediate position. Its main text studies a semi-finite slab or strip geometry, finite along \(x\) and translationally invariant along \(y\). In this geometry, the system supports ordinary gap edge states and, in a regime where both media have the same nontrivial topology but different rotation parameters, two strongly localized edge states that coexist with gapless bands while remaining unhybridized with bulk extended states. These are identified as dispersive TBICs. In a fully finite geometry, however, the same states become corner states rather than remaining extended along an edge, so the evidence is specific to the semi-infinite strip setup [2503.17263].

## 4. Topological characterization and spectral embedding

Surface TBICs are characterized by the same bulk-boundary logic that governs ordinary topological boundary states, but with the added requirement that the boundary dispersion remain embedded rather than isolated. In the phononic-crystal realization, the upper gap of the \(h^{(3)}\) subsystem carries a \(k_z\)-dependent valley Chern number
$$
C^{(3)}(k_z)=\frac{1}{2}
$$
along the \(KH\) line, identifying \(h^{(3)}\) as a 3D valley topological insulator. The top surface states themselves possess a surface valley topology with
$$
C_R=\frac{1}{2},
$$
and the system also has nonzero \(z\)-direction Zak phases \(\theta_z(k_x,k_y)\), which generate top-surface boundary states [2509.00344]. The 2D interface TBICs then arise from the 3D valley topology of \(h^{(3)}\), while the 1D hinge TBICs arise from the 2D valley topology of the top-surface states.

In the mirror-stacked framework, topological characterization is inherited rather than recomputed. Because \(h_{\mathrm{even}}=h-t_c I\) and \(h_{\mathrm{odd}}=h+t_c I\), each sector preserves the monolayer invariant exactly. For the SSH example, the relevant condition is \(t_1/t_0>1\), corresponding to a quantized dipole moment and a topological boundary state. For the quadrupole model, the same inequality yields a quantized quadrupole moment and corner states. The TBIC transition occurs when the shifted boundary-state energy enters the opposite-parity continuum, not when the topological invariant changes [2212.06562].

In the colored quantum random walk, topology is encoded in three Floquet quasienergy bands with Chern numbers computed from the Berry curvature. The phase diagram contains trivial and nontrivial sectors, including band triplets such as \((1,-1,0)\) and \((0,-1,1)\). The quasienergy spectrum is defined in the Floquet “energy Brillouin zone”
$$
-\pi < E(k_x,k_y) < \pi.
$$
The reported TBICs are not gap states: they are embedded in the quasienergy range of extended bulk states, and their appearance is tied to color-induced band mixing and topological Floquet structure rather than to a closed-form decoupling rule [2503.17263].

## 5. Experimental signatures, robustness, and diagnostics

The decisive signature of a surface TBIC is simultaneous spectral embedding and spatial localization. In the phononic-crystal experiment, the operating gap of the \(h^{(3)}\) subsystem is \(9.50\ \text{kHz} - 9.75\ \text{kHz}\), and a representative field map at \(f=9.68\ \text{kHz}\) shows strong localization of acoustic pressure at the \(XZ\) interface. A 2D Fourier transform of the measured pressure field yields isofrequency contours that match the simulated interface-TBIC band, and projected dispersions are measured along \(k_x\) at \(k_z=0.5\pi/h\) and along \(k_z\) at \(k_x=0.5\pi/a_0\). The paper emphasizes that no signal of the overlapped bulk states is observed in these measurements, despite spectral overlap with the continuum [2509.00344].

Robustness in that system is evaluated against valley-preserving defects and geometric size errors. Supercell calculations with defect cavities placed on the interface, aligned along the interface centerline, or randomly distributed on the interface preserve the interface-TBIC dispersion without hybridization, and transport simulations show propagation along the intended paths without bulk leakage. The same qualitative robustness is reported for modified cavity diameters, again with interface localization maintained [2509.00344].

The mirror-stacked acoustic experiments establish another important diagnostic: sector-selective excitation. In the SSH bilayer, inphase excitation accesses the even mirror sector and antiphase excitation accesses the odd sector. At \(f_1\approx 6.90\ \text{kHz}\), inphase driving produces an edge-localized even-parity state while antiphase driving excites an extended odd-parity bulk state; at \(f_2\approx 7.45\ \text{kHz}\), the parity roles reverse. This directly exhibits a localized state whose frequency lies inside the bulk-band window of the opposite parity sector [2212.06562].

The Floquet cQRW work presents a methodological caveat. It identifies “strongly localized edge states” and marks dispersive TBICs in strip spectra, but the main text does not provide explicit inverse participation ratios, penetration depths, or spatial probability profiles of the starred states. The surface character is therefore supported primarily by strip-spectrum analysis and descriptive identification rather than by exhaustive localization diagnostics [2503.17263]. This has become a standard caution in the field: spectral embedding alone is insufficient, and definitive surface-TBIC characterization benefits from direct real-space weight profiles and finite-size scaling.

## 6. Relation to broader surface-state theory and open problems

Surface TBICs are a special subclass of topological boundary states, and the broader theory of topological surfaces remains essential for interpreting them. Space-group classification already distinguishes \(\Gamma\)-states, translationally-active states, and valley topological insulators, with boundary robustness depending on whether protection comes from time-reversal symmetry alone or from crystalline symmetry as well [1209.2610]. Separate semi-infinite tight-binding calculations for \(\mathrm{Bi}_{1-x}\mathrm{Sb}_x\) show that even ordinary topological surface states depend strongly on surface orientation and termination: metallicity is protected, but dispersion, connectivity, and which surface TRIM are enclosed by Fermi contours can change substantially when the boundary geometry changes [1402.5751]. This suggests that surface TBIC dispersions and termination dependence should also be geometry sensitive, although the present TBIC-specific literature has not yet produced a comparably systematic survey.

Several open issues follow directly from the current record. First, direct 3D surface-TBIC demonstrations are still sparse; the strongest explicit experimental case is the 3D phononic-crystal interface realization [2509.00344]. Second, universal design principles exist, but explicit 3D constructions remain underdeveloped: the mirror-stacking framework is formulated for arbitrary monolayer models and strongly implies extension to genuine surface TBICs, yet the paper itself demonstrates only 1D first-order and 2D higher-order cases [2212.06562]. Third, some claimed edge or surface TBICs are supported more strongly by spectral evidence than by localization diagnostics, as in the cQRW strip geometry [2503.17263].

A persistent misconception is that any topological surface state embedded in a projected band structure is automatically a TBIC. The literature indicates a stricter criterion. A TBIC must remain localized while spectrally embedded, and that requires a mechanism forbidding hybridization with the relevant continuum. In current realizations, that mechanism is supplied by separability, mirror parity, or sector orthogonality rather than by topology alone [2509.00344]. Another misconception is that TBICs must be nondispersive. Both the phononic-crystal surface TBICs and the cQRW edge TBICs are explicitly dispersive, with propagation characteristics inherited from valley topology or Floquet band structure [2509.00344][2503.17263].

Surface TBICs therefore occupy a precise niche at the intersection of bulk topology, boundary-state theory, and continuum decoupling. Their defining achievement is not merely the creation of a protected boundary mode, but the maintenance of that mode as a confined propagating state inside a spectrum that should, in ordinary circumstances, delocalize it.

Source: https://www.emergentmind.com/topics/surface-tbics