Edge-Selective Bulk-Boundary Correspondence

• Edge-selective bulk-boundary correspondence refines traditional bulk-edge links by incorporating termination sensitivity and probe-specific boundary characteristics.
• Methodologies include spectral flow, winding number analysis, and self-adjoint extension frameworks to elucidate edge state behavior under varied conditions.
• The framework advances understanding of higher-order topology and non-Hermitian corrections, offering precise insights into edge-specific invariants and boundary phenomena.

Edge-selective bulk-boundary correspondence denotes a family of refinements of bulk-boundary correspondence in which bulk topology is reflected not merely by the existence of boundary modes, but by boundary-resolved, termination-sensitive, or probe-selective structure. The phrase is used explicitly for BDI-class higher-order topological insulators in which breaking mirror symmetries that anticommute with the chiral operator produces gap closings and bound states only along a subset of boundaries of the same orientation (Aich et al., 26 Sep 2025). Closely related constructions appear as signed left/right edge events in topological pumping, termination-adapted sublattice invariants in trimer Su-Schrieffer-Heeger chains, geometric multiplicity factors for curved-interface conductance measurements, and boundary-condition corrections carried by self-adjoint extension data (Imura et al., 2017, Anastasiadis et al., 2022, Drouot et al., 2024, Kellendonk et al., 1 Oct 2025).

1. Meanings and scope of selectivity

Across the literature, “selective” does not denote one universal mechanism. In some settings the relevant selectivity is event-wise and edge-resolved: individual left-edge and right-edge spectral-flow events contribute with different signs, but their signed sum reproduces a single bulk invariant. In others, the selectivity is tied to how the sample is cut, to which boundary region is probed, to which boundary condition is imposed, or to whether distinct parallel edges remain symmetry-equivalent after symmetry breaking. Several papers are explicit that these meanings should not be conflated: non-Bloch non-Hermitian correspondence is boundary-sensitive without assigning separate left-edge and right-edge invariants, bulk-LDOS correspondence is selective by boundary dimension rather than by edge identity, and finite-sample photonic correspondence is a global circulation law rather than an edge-by-edge decomposition (Xiao et al., 2019, Xie et al., 2022, Qiu et al., 26 Jan 2025).

Setting	Selective boundary object	Bulk counterpart
1D topological pump	signed center-of-mass jumps from left/right edge crossings	pumped charge / Chern number
SSH3 chain	termination-adapted normalized sublattice Zak’s phase	change in edge-state count from a given band
Non-Hermitian Floquet walk	non-Bloch invariants on generalized Brillouin zones	existence of topological edge states
BDI higher-order insulator	edge-sensitive winding $\nu_1^{\mathcal c_2}$	selected edge gap closing and surviving corner states
Curved interface	geometric integer $X_{U,V}$	Hall-conductance difference
Finite photonic structure	global boundary-circulation index $EI_L(\Delta)$	gap Chern number

A persistent distinction therefore runs through the subject. Some works establish only a boundary-sensitive or boundary-condition-dependent correspondence; others isolate a genuinely edge-selective mechanism in which different edges of the same orientation are topologically distinguished. The strongest explicit statement of the latter type is the selective higher-order mechanism based on anticommuting mirror and chiral symmetries (Aich et al., 26 Sep 2025).

2. Spectral flow, pumping, and one-edge winding formulations

In the 1D topological pump, bulk-edge correspondence can be written as an edge-resolved accounting identity rather than as a static “edge-state existence” statement. For a finite open chain, the center of mass $\bar x(t)$ separates into continuous adiabatic bulk drift and discontinuous jumps when an edge level crosses the Fermi energy. The bulk transported charge is

$$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$

and the edge quantity satisfies

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$

At the level of individual events, the jump is

$$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$

This resolves the correspondence by edge side and by whether an edge state enters or leaves the occupied subspace. The construction is selective in the sense of a temporally ordered sum of left-edge and right-edge events, but it does not define independent left and right topological indices; the signed sum reproduces the bulk Chern number (Imura et al., 2017).

The same paper also isolates a useful robustness principle. In the presence of disorder, impurity-localized states in the gap can generate additional, generally non-quantized center-of-mass jumps,

$$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$

but each impurity crossing appears twice with opposite slope sign, so the pair cancels over a cycle. By contrast, edge jumps evade pairwise cancellation because of the extra factor ${\rm sgn}(x_{\rm edge})$. The topological correspondence is therefore not attached to every localized crossing event; it is attached specifically to boundary-localized spectral flow (Imura et al., 2017).

A different one-edge spectral-flow formulation appears in Sturmian and Kohmoto-like quasiperiodic chains. There the relevant bulk invariant is the gap label $N+n\theta$, while the edge invariant is a winding number of half-space Dirichlet eigenvalues defined only after passing from the genuine Sturmian hull to an augmented phason cycle. The correspondence theorem reads

$X_{U,V}$0

This construction is edge-resolved in the strict half-space sense: the invariant is attached to a chosen compression $X_{U,V}$1 on $X_{U,V}$2, not to both ends simultaneously. It also shows that in quasicrystals the existence of a meaningful edge winding can require augmentation or smoothing of the phason dynamics (Kellendonk et al., 2017).

3. Termination-sensitive and boundary-profile-dependent formulations

The trimer Su-Schrieffer-Heeger model makes edge selectivity explicit at the level of chain termination. In SSH3, the ordinary Zak phase is quantized only for a mirror-symmetric chain and fails as a robust invariant once mirror symmetry is lost, even though localized edge states persist. The replacement is the normalized sublattice Zak’s phase, defined by projecting the Bloch eigenvector onto a selected sublattice component in a specified gauge. For a $X_{U,V}$3 chain the relevant quantity is $X_{U,V}$4; for $X_{U,V}$5 it is $X_{U,V}$6; for $X_{U,V}$7 the formal invariant $X_{U,V}$8 vanishes trivially. The physical content is termination-selective and band-selective: the gauge-invariant difference between target and reference normalized sublattice Zak phases, divided by $X_{U,V}$9, equals the change in the number of edge states emerging from band $EI_L(\Delta)$0 for the chosen cut. The same bulk crystal therefore requires different sublattice-resolved invariants depending on how the boundary is realized (Anastasiadis et al., 2022).

A broader boundary-profile-dependent formulation was developed for band insulators using Green-function poles. One starts from a periodic bulk Hamiltonian $EI_L(\Delta)$1, introduces a local boundary operator $EI_L(\Delta)$2, and interpolates through

$EI_L(\Delta)$3

The poles entering the gap are encoded by

$EI_L(\Delta)$4

The resulting bulk numbers are computed from bulk eigenstates, but they are explicitly sensitive to the chosen edge, open boundary, or grain boundary because that information is encoded in $EI_L(\Delta)$5. In two dimensions, the pole winding number on the $EI_L(\Delta)$6 cylinder distinguishes robust metallic edge bands from trivial pole curves. This is a bulk-based but boundary-profile-aware correspondence rather than a boundary-blind one (Rhim et al., 2017).

For nearest-layer Dirac Hamiltonians, edge dependence appears geometrically. Fixing the conserved momentum $EI_L(\Delta)$7 and treating the transverse momentum as $EI_L(\Delta)$8, the bulk vector traces an ellipse

$EI_L(\Delta)$9

An edge state exists if and only if the projected ellipse $\bar x(t)$0 encloses the origin, and its energy is

$\bar x(t)$1

In the $\bar x(t)$2 case,

$\bar x(t)$3

This makes the detailed boundary spectrum strongly selective to the chosen edge orientation or termination, while the net chiral spectral flow remains fixed by the bulk Chern number. The detailed edge dispersion is therefore boundary-selective even when the topological count is not (Mong et al., 2010).

4. Non-Hermitian generalizations and anomalous counting

Non-Hermitian systems sharpen the difference between boundary sensitivity and genuine edge selectivity. In non-unitary discrete-time quantum walks of single photons, ordinary Bloch invariants fail because bulk eigenstates under open or domain-wall conditions become exponentially localized at boundaries by the non-Hermitian skin effect. The correspondence is restored only after replacing Bloch invariants by non-Bloch invariants computed on generalized Brillouin zones derived from localized bulk-state wavefunctions. At the same time, the paper demonstrates photon localization near boundaries even in the absence of topological edge states, so boundary localization by itself is not a topological diagnostic (Xiao et al., 2019).

A complementary route replaces the non-Hermitian eigendecomposition by singular-value decomposition. Writing

$\bar x(t)$4

the unitary $\bar x(t)$5 plays the role of a flattened Hamiltonian. In Hermitian systems this reduces to the usual flattening, while in non-Hermitian systems it avoids the spectral instability of complex eigenvalues. Restricting $\bar x(t)$6 to a subsystem $\bar x(t)$7 yields a generalized entanglement-spectrum construction whose singular spectrum reproduces the open-boundary edge physics. In this framework, the periodic bulk Hamiltonian can again determine the edge structure, but only after the relevant topological data have been transferred from complex eigenvalues to singular values (Herviou et al., 2018).

Anomalous edge counting requires an additional distinction. In the non-Hermitian SSH model, the paper on anomalous bulk-boundary correspondence defines the conventional finite-chain count as

$\bar x(t)$8

and the anomalous regime by

$\bar x(t)$9

The defective edge state is then interpreted as a consequence of a boundary exceptional point of an effective edge Hamiltonian. This separates non-Bloch BBC, which corrects the bulk phase diagram, from a boundary anomaly that changes the number of linearly independent edge states and can localize a state only on the left edge or only on the right edge (Wang et al., 2019).

A many-body diagnostic of the same issue is edge entanglement entropy in the non-Hermitian SSH-Hubbard chain,

$$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$0

For the free model, the topological transition inferred from open boundaries is at $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$1, while the periodic Bloch gap closings are at $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$2. The edge entanglement entropy identifies all three special points, becomes complex in the point-gap phases, and remains useful after introducing Hubbard interaction. The interaction progressively suppresses the mismatch between open- and periodic-boundary diagnostics and eventually restores a Hermitian-like correspondence (Chen et al., 2021).

5. Higher-order topology and genuinely selective edges

The strongest explicit realization of edge-selective bulk-boundary correspondence appears in BDI-class higher-order topological insulators with chiral and mirror symmetries. For models of the form

$$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$3

the two physical mirrors $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$4 and the chiral operator $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$5 anticommute pairwise. In the unperturbed tensor-product model, the higher-order invariant is

$$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$6

which gives $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$7 zero modes per corner. Breaking a mirror that anticommutes with $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$8, while preserving $$\Delta \bar{x}_{\rm net}=C(n_F),\qquad C(n_F)=\sum_{n=1}^{n_F} C_n,$$9 itself, yields a boundary transition rather than a bulk one. For the perturbation $I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$0, projection onto the edge sector produces

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$1

and the new edge-sensitive invariant is

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$2

This construction distinguishes opposite edges of the same orientation because they carry opposite $I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$3 labels. For $I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$4, the selective transition to the trivial phase occurs on the edge with $I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$5 at

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$6

After the transition, only two of the four corners retain zero modes. This is a genuinely topological selection among parallel edges, rather than a mere redistribution of boundary weight (Aich et al., 26 Sep 2025).

A closely related but more general boundary-centered mechanism is the edge-network theory for higher-order topology and topological fullerenes. There the effective boundary Hamiltonian is

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$7

with

$I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$8

Localized bound states arise only where adjacent boundary masses are inequivalent. At a corner between edges $I(n_F)=C(n_F),\qquad I(n_F)-I(n_F-1)=C_{n_F}.$9 and $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$0, a bound state exists when $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$1, with

$$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$2

In this formulation the bulk does not merely imply “some boundary state somewhere”; it induces a patterned set of edge masses, and only those corners, hinges, or multi-leg junctions where the mass texture develops a topological defect host localized modes (Wang et al., 2018).

Taken together, these higher-order constructions show that edge selectivity can be stronger than codimension selectivity. It can distinguish not only edge versus corner, but also one parallel edge from another, or one corner and junction from another, through symmetry-controlled boundary mass data (Aich et al., 26 Sep 2025, Wang et al., 2018).

6. Finite geometries, curved interfaces, and dimensional selectivity

For curved interfaces in Hall insulators, selectivity enters through the measurement region rather than through a boundary-local invariant assigned to each microscopic edge mode. If $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$3 interpolates between two bulks $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$4 and $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$5 across an interface $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$6, and current is measured into a set $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$7, then

$$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$8

The integer $$\Delta \bar{x}=\frac12\,{\rm sgn}(x_{\rm edge})\,[-{\rm sgn}({\rm slope})].$$9 depends only on the geometry of $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$0 and $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$1 and counts the algebraic number of times that the oriented interface enters the measurement set. The correspondence is therefore selective in a probe-based geometric sense: changing $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$2 changes which portions of a complicated boundary contribute to the measured conductance (Drouot et al., 2024).

In finite photonic structures, the natural boundary invariant is global rather than edge-by-edge. For the divergence-form operator restricted to a bounded domain $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$3 with Dirichlet boundary conditions, the edge index is

$$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$4

and the finite-geometry bulk-edge correspondence is

$$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$5

The paper interprets $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$6 as a circulation of electromagnetic energy along the full boundary of the sample. This yields a rigorous finite-sample correspondence, but not an explicit decomposition into separate sides, arcs, or corners (Qiu et al., 26 Jan 2025).

A weaker but practically important selectivity appears in the bulk-LDOS correspondence of finite topological photonic crystals. There the local-density of states is partitioned among the two-dimensional interior bulk area, the one-dimensional edge region, and zero-dimensional corner sites. The region-averaged observables are

$$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$7

with $$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$8. This is hierarchical boundary selectivity by dimension or codimension, not by individual edge identity. The paper is explicit that it does not establish a left-edge/right-edge or orientation-resolved correspondence (Xie et al., 2022).

These works delimit an important conceptual boundary. Selectivity may refer to a chosen probe region, to a global circulation along a closed boundary, or to a bulk/edge/corner hierarchy. None of these by itself implies the stronger same-orientation edge discrimination realized in higher-order chiral-mirror systems (Drouot et al., 2024, Qiu et al., 26 Jan 2025, Xie et al., 2022).

7. Boundary conditions, self-adjoint extensions, and corrected correspondence

Continuum models make boundary selectivity explicit because the same bulk differential expression can admit inequivalent self-adjoint edge realizations. In the odd-viscous rotating shallow-water model, the bulk Chern number of the upper band is fixed,

$$\Delta \bar{x}=x_{\rm imp}\,[-{\rm sgn}({\rm slope})],$$9

but the edge index

${\rm sgn}(x_{\rm edge})$0

depends strongly on the chosen local self-adjoint boundary condition. The paper classifies all such boundary conditions into four families, DD, ND, DN, and NN, distinguished in part by ${\rm sgn}(x_{\rm edge})$1, the number of normal-derivative degrees of freedom constrained at the boundary. Bulk-edge correspondence in this setting means

${\rm sgn}(x_{\rm edge})$2

and violation means

${\rm sgn}(x_{\rm edge})$3

Within families, both correspondence and violation are typical, and the maximally possible amount of violation increases with the degree of the boundary operator. The paper also identifies a single spectral mechanism for changing ${\rm sgn}(x_{\rm edge})$4: a transition between asymptotically parabolic and asymptotically flat edge branches (Graf et al., 2024).

A general ${\rm sgn}(x_{\rm edge})$5-theoretic account of this phenomenon is provided by the theory of self-adjoint extensions. For a symmetric operator ${\rm sgn}(x_{\rm edge})$6, self-adjoint extensions are parametrized by von Neumann unitaries ${\rm sgn}(x_{\rm edge})$7, and the resolvent difference satisfies

${\rm sgn}(x_{\rm edge})$8

If ${\rm sgn}(x_{\rm edge})$9 and $N+n\theta$0 are two self-adjoint extensions of the same symmetric operator, then their relative edge class is

$N+n\theta$1

Under strong affiliation assumptions the standard bulk-edge theorem survives,

$N+n\theta$2

but more generally the correct statement is relative or corrected by boundary data. In the half-space setting this takes the form

$N+n\theta$3

Apparent violations of the classical bulk-edge paradigm are thereby reinterpreted as boundary-induced topological corrections rather than as a breakdown of topology itself (Kellendonk et al., 1 Oct 2025).

This perspective clarifies a central misconception. Edge-selective behavior does not necessarily mean that bulk topology has failed to constrain the boundary. It may instead mean that the relevant topological statement is relative to a chosen edge realization, or that an additional boundary $N+n\theta$4-class must be included. In continuum settings, selectivity can therefore be intrinsic to the topology of boundary conditions themselves (Graf et al., 2024, Kellendonk et al., 1 Oct 2025).