Boundary Zero Singular Modes (ZSMs)

• Boundary Zero Singular Modes (ZSMs) are boundary-localized states pinned to zero energy by topological invariants, bulk singularities, and non-Hermitian effects.
• They emerge in diverse systems such as flatband lattices, singular Sturm–Liouville operators, and non-Hermitian chains where boundary defects or discontinuities enforce their existence.
• Experimental realizations in photonic lattices and PT-symmetric chains demonstrate their robustness and provide key benchmarks for bulk-boundary correspondence.

Boundary Zero Singular Modes (ZSMs) are a broad class of boundary-localized states, generically pinned to zero energy (or, more generally, spectral singularities), whose existence is enforced either by nontrivial topology, singularities in bulk wavefunctions, non-Hermitian or nonnormal matrix structure, or by the introduction of boundary defects, holes, or singular endpoints. ZSMs appear in a wide range of systems, including flatband lattices, singular Sturm–Liouville operators, 1D topological chains, and topological field theories. Their unifying attribute is that they are enforced by global, singular, or boundary-induced properties—not merely local or symmetry-based criteria—and their counting, robustness, and mathematical signatures provide key benchmarks for bulk-boundary correspondence in both Hermitian and non-Hermitian systems.

1. General Definition and Classification

ZSMs are defined by the vanishing of specific spectral quantities at the boundary—eigenvalues or singular values—under open (as opposed to closed) boundary conditions. In flatband systems, singularities in the bulk Bloch wavefunction (e.g., discontinuities at high-symmetry points) force the existence of boundary or "robust boundary modes" at zero energy. In non-Hermitian or nonnormal systems, ZSMs are more properly characterized by the vanishing of certain singular values, leading to long-lived or even strictly zero-energy boundary-localized states, even in the absence of global symmetries or bulk topological indices. In continuous models, such as singular Sturm–Liouville operators, a ZSM is any normalizable solution that both vanishes at a singular endpoint and satisfies the natural boundary condition (e.g., Neumann).

ZSMs may be driven by:

• Bulk singularities (irreparable discontinuities in the spectrum or wavefunctions)
• Topological invariants (e.g., winding numbers, homology Betti numbers)
• Boundary defects, holes, impurities, or non-Hermitian attachments
• Nonnormality and pseudospectra structure of the Hamiltonian

2. ZSMs in Singular Flatband Lattices

In paradigmatic tight-binding models with exactly flat bands (such as the Kagome lattice), the flatband eigenvector,

$$\psi_{\rm FB}(k)\;=\; \frac{1}{a_k} \begin{pmatrix} e^{i k\cdot a_1}-1\ 1-e^{i k\cdot(a_1-a_2)}\ e^{i k\cdot(a_1-a_2)}-e^{i k\cdot a_1} \end{pmatrix},\quad a_k = \sqrt{2\bigl[3-\cos(k\cdot a_1)-\cos(k\cdot a_2)-\cos(k\cdot(a_1-a_2))\bigr]}$$

suffers a singularity at $k=0$ due to vanishing numerator, which signals that a global basis of compact localized states (CLSs) cannot be constructed—an immovable discontinuity. As a result, when holes (inner boundaries) are introduced (by site removal or strong detuning), inner robust boundary modes (RBMs)—the canonical ZSMs in this context—form along the hole perimeters. Their counting is determined by the bulk–hole correspondence: $$N_{\rm flat} = N_{\rm CLS} + N_{\rm RBM},\qquad N_{\rm RBM} = \beta_1 = N_{\rm holes}$$ where $\beta_1$ is the first homology (Betti number). The existence of one RBM per hole is topologically protected, independent of the hole shape or lattice size.

These ZSMs were directly observed in laser-written Kagome photonic lattices, where the number and robustness of inner RBMs were demonstrated by precisely matching wavefunction phases to the boundary sites around each hole (Song et al., 3 Dec 2024).

3. ZSMs in Singular Sturm–Liouville Theory

For differential operators on intervals where coefficient functions or the domain render an endpoint singular (e.g., infinite interval, vanishing $r(x)$ or $w(x)$), a boundary ZSM is any eigenfunction satisfying:

• Homogeneous Neumann boundary condition at the singular endpoint,
• Vanishing as $x\to a^+$,
• Normalizability and discreteness under the self-adjoint extension.

A practical sufficient condition for the existence of a ZSM at $x=a$ is: $\lim_{x\to a^+} \frac{|w'(x)|}{w(x)} = \infty$ when $r/w$ is bounded and of bounded variation. Under these criteria, there exists exactly one normalized eigenfunction vanishing at the endpoint (the ZSM), and the spectrum remains discrete and bounded below. The classic example is the Hermite functions on $\mathbb{R}$, which vanish at both infinities and form the unique ZSMs for those singular endpoints (Richthofer et al., 2020).

4. Non-Hermitian and Topological Lattice ZSMs

In non-Hermitian lattice systems, conventional bulk-boundary correspondence (i.e., relating zero eigenvalues under OBC to a bulk winding number) fails: the skin effect and sensitivity to boundary conditions generically hide or shift edge-localized zero modes. The correct correspondence is restored at the singular-value (SVD) level: Boundary Zero Singular Modes correspond to vectors $v$ with singular value $s \to 0$ as $L \to \infty$. For a block Toeplitz Hamiltonian $H_L$, the number of ZSMs is determined by the sum of the winding numbers of $H$ and its reflected matrix (Monkman et al., 15 May 2024), i.e.,

$K = \dim\ker H + \dim\ker \tilde H,$

with at least $\lvert W[H] \rvert$ ZSMs guaranteed by the nonzero point-gap winding $W[H]$. In multiband cases, this count may be a lower bound due to additional hidden or accidental ZSMs.

In practice, these ZSMs manifest as exponentially localized boundary states, whose singular values decrease as $s_n(L) \sim \exp(-L/\xi)$, yielding extremely long-lived, zero-energy boundary excitations, observable (e.g.) via Loschmidt echo or quench protocols in photonic or cold-atom platforms.

5. Nonnormality-Protected and Symmetry-Free ZSMs

Beyond topology, ZSMs can arise purely due to nonnormality—the Hamiltonian's non-commutation with its adjoint—resulting in boundary-localized singular resonances even absent bulk gap or quantized invariants. Pseudospectrum theory quantifies the robustness and existence of such modes: given a (possibly Hermitian) Hamiltonian $H$, ZSMs are present whenever there exist boundary-localized pseudo-eigenvectors at zero energy with $E \in \Lambda_\epsilon(H)$, the $\epsilon$-pseudospectrum. Here, strong nonnormality (measured by the operator condition number or commutator norm) ensures that boundary perturbations or disorder leave the ZSM robust up to exponentially small splittings (Okuma et al., 2020).

In non-Hermitian settings, symmetry-free ZSMs (SFZMs) can be deterministically nucleated at boundaries by constructing a non-Hermitian "nucleus" attached to a bulk, with carefully chosen couplings enforcing the singular mode precisely at zero energy, robust to arbitrary bulk disorder and topology (Rivero et al., 2023).

6. ZSMs in Field Theory, Topological Chains, and Beyond

Boundary ZSMs appear as physical or algebraic engines in field-theoretic and highly correlated systems:

• In infinite-component $BF$ topological field theories, ZSMs of Toeplitz $K$ matrices mediate nontrivial boundary-to-boundary braiding (Toeplitz braiding), responsible for topological phases in fracton orders and non-Hermitian amplification, with explicit analytic and numerical correspondence demonstrated for Hatano–Nelson- and SSH-type cases (Li et al., 12 Nov 2025).
• In strongly interacting 1D quantum chains (e.g., Kitaev, Ising/QIC, sine-Gordon/massive Thirring, $\mathbb{Z}_N$ parafermion and non-Abelian dyonic chains), ZSMs appear as protected boundary modes responsible for zero-energy ground state degeneracy, nonlocal Majorana or parafermion operators, and serve as precise endpoints for topological manipulations and fusion rules (Müller et al., 2016, Pasnoori et al., 18 Mar 2025, Munk et al., 2018).
• The explicit construction and manipulation of ZSMs are central to the design of robust platforms for topological quantum information processing.

7. Experimental Realizations and Robustness

ZSMs have been directly observed in:

• Photonic Kagome lattices with engineered holes, where the number and topological nature of inner RBMs (ZSMs) coincide with the Betti number and match theoretical predictions (Song et al., 3 Dec 2024).
• PT-symmetric SSH and Kitaev chains, where gain/loss potentials can restore or pin real zero modes at finite size by decoupling hybridizations via imaginary energy, visible as characteristic eigenfrequency signals (Xu et al., 2019, Sakaguchi et al., 2022).
• Hermitian and non-Hermitian systems with nonnormal or synthetic boundary elements, via boundary lasing, singular value analysis, or topological charge measurements.

Robustness of ZSMs, even in the presence of disorder, topological triviality, or spectral instabilities, is anchored by the global, singular or topological criteria (e.g., the enforcement of boundary singularities, the presence of essential singularities in bulk wavefunctions, or the satisfaction of nonnormality-based decoupling constraints). In continuous cases, unique ZSMs exist under mild regularity or blow-up conditions at singular endpoints (Richthofer et al., 2020). In lattice models, boundary nucleation of ZSMs may be tailored via boundary engineering and nonreciprocal coupling architectures.

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References:

• "Bulk-hole correspondence and inner robust boundary modes in singular flatband lattices" (Song et al., 3 Dec 2024)
• "Singular Sturm-Liouville Problems with Zero Potential (q=0) and Singular Slow Feature Analysis" (Richthofer et al., 2020)
• "Hidden zero modes and topology of multiband non-Hermitian systems" (Monkman et al., 15 May 2024)
• "Robust zero modes in non-Hermitian systems without global symmetries" (Rivero et al., 2023)
• "Hermitian zero modes protected by nonnormality: Application of pseudospectra" (Okuma et al., 2020)
• "Bulk--Boundary Correspondence and Boundary Zero Modes in a Non-Hermitian Kitaev Chain Model" (Sakaguchi et al., 2022)
• "Fate of zero modes in a finite Su-Schrieffer-Heeger Model with $\mathcal{PT}$ Symmetry" (Xu et al., 2019)
• "Infinite-component $BF$ field theory: Nexus of fracton order, Toeplitz braiding, and non-Hermitian amplification" (Li et al., 12 Nov 2025)
• "Classical impurities and boundary Majorana zero modes in quantum chains" (Müller et al., 2016)
• "Duality symmetry, zero energy modes and boundary spectrum of the sine-Gordon/massive Thirring model" (Pasnoori et al., 18 Mar 2025)
• "Dyonic zero-energy modes" (Munk et al., 2018)