Singular Flat Bands & Topological Discontinuities

• Singular flat bands are defined by an immovable Bloch wave discontinuity at a band crossing, leading to an incomplete basis of compact localized states.
• The construction of such bands uses Fourier-transformed compact localized states where zeros enforce the need for additional extended states like non-contractible loop states.
• Quantum geometry and interaction effects in singular flat bands drive phenomena such as anomalous Landau-level spreading and enhanced superconductivity.

Singular flat bands are flat bands whose Bloch wave functions possess an immovable discontinuity generated by a band crossing with another band. In this class, the flat-band eigenstate is ill-defined at the touching momentum, the associated vector bundle cannot be defined over the entire Brillouin zone, and translated compact localized states do not form a complete basis; additional extended states are required to span the flat-band eigenspace. This distinguishes singular flat bands from nonsingular flat bands, whose Bloch eigenfunctions remain analytic and whose eigenmode space can be completely spanned by compact localized states (Rhim et al., 2018, Rhim et al., 2020).

1. Definition and classification

The basic classification criterion is the singular behavior of the flat-band Bloch wave function in momentum space. A singular flat band has an immovable discontinuity at a momentum point $\mathbf{k}_0$, typically induced by a band touching with a dispersive band. A nonsingular flat band has no such discontinuity, even if a degeneracy is present. In the formulation of Rhim and Yang, a flat band is nonsingular if there exists an analytic function $\alpha_{\mathbf{k}}$ such that $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$ is analytic throughout the Brillouin zone; it is singular if for any choice of $\alpha_{\mathbf{k}}$, $\alpha_{\mathbf{k}}$ must vanish at some $\mathbf{k}_0$, making the normalized eigenstate ill-defined there (Rhim et al., 2018).

This distinction has immediate structural consequences. In a nonsingular flat band, a complete and linearly independent set of $N$ translated compact localized states can span the band. In a singular flat band, the set of compact localized states is incomplete, and extended states such as non-contractible loop states must supplement it. The singularity is generated by a band crossing, so isolated flat bands are analytic everywhere; conversely, lifting a singular touching generally destroys exact flatness and can produce a nearly flat band with finite Chern number, whereas lifting a nonsingular touching can preserve perfect flatness (Rhim et al., 2018).

The review literature makes the same separation in a complementary language: singular flat bands are defined by discontinuous or ill-defined normalized Bloch eigenstates at isolated band-degeneracy points, while nonsingular flat bands remain analytic across the Brillouin zone. In this usage, all one-dimensional flat bands belong to the nonsingular class. A common misconception is therefore that any band touching makes a flat band singular; the classification depends on the wave-function singularity, not on degeneracy alone (Rhim et al., 2020).

2. Construction from compact localized states and FT-CLS singularity

A systematic construction of singular and nonsingular flat-band models starts from a compact localized state (CLS), its symmetry representation, and the singularity structure of the Fourier-transformed CLS (FT-CLS). The FT-CLS is the momentum-space wave function

$$\hat{u}(\mathbf{k}) = \left( \sum_{\mathbf{R}} S_1(\mathbf{R}) e^{i \mathbf{k}\cdot\mathbf{R}}, \ldots, \sum_{\mathbf{R}} S_N(\mathbf{R}) e^{i \mathbf{k}\cdot\mathbf{R}} \right),$$

and the criterion is direct: if $\hat{u}(\mathbf{k}_*)=0$ at some momentum $\mathbf{k}_*$, the resulting flat band is singular and has a symmetry-enforced band crossing at that point; if $\alpha_{\mathbf{k}}$0 is nonzero throughout the Brillouin zone, the band is nonsingular (Hwang et al., 2021).

The constructive procedure is to choose a CLS with a desired symmetry representation, determine whether the FT-CLS is singular, construct basis molecular orbitals orthogonal to the FT-CLS, and assemble a tight-binding Hamiltonian from their outer products. In the notation of the construction,

$\alpha_{\mathbf{k}}$1

with $\alpha_{\mathbf{k}}$2 Laurent polynomials and $\alpha_{\mathbf{k}}$3. This guarantees a flat band at zero energy and, through the singularity of $\alpha_{\mathbf{k}}$4, fixes whether the band is singular or nonsingular. The Lieb lattice flat band with an FT-CLS zero at the $\alpha_{\mathbf{k}}$5 point and the kagome lattice flat band with an FT-CLS zero at $\alpha_{\mathbf{k}}$6 are canonical singular examples in this framework, while modified CLS choices can produce nonsingular flat bands on the same underlying lattice (Hwang et al., 2021).

The same formalism also clarifies why singularity is not an accidental feature of fine-tuned hopping amplitudes. The singular or nonsingular character is encoded in the spatial shape and symmetry of the CLS and in the zeros of the FT-CLS. A plausible implication is that singularity is best viewed as a wave-function constraint rather than a merely spectral one.

3. Real-space topology, loop states, and boundary modes

The incompleteness of compact localized states in a singular flat band has a real-space topological manifestation. On periodic geometries, singular flat bands require non-contractible loop states (NLSs) in two dimensions, or more generally extended non-compact states, in order to span the flat-band subspace. On open geometries, these become robust boundary modes. The review literature describes this as a novel bulk-boundary correspondence: the robust boundary mode is guaranteed by the singularity of the Bloch wave function, even though the flat band is topologically trivial in the conventional Berry-curvature or Chern-number sense (Rhim et al., 2020).

A useful geometric diagnostic is the Hilbert-Schmidt or quantum distance between nearby Bloch eigenstates,

$\alpha_{\mathbf{k}}$7

For nonsingular flat bands, this distance vanishes as $\alpha_{\mathbf{k}}$8. For singular flat bands it can remain finite at the touching point. In tunable thin-plate acoustic metamaterials, the kagome flat band is singular because the Hilbert-Schmidt distance between eigenstates at $\alpha_{\mathbf{k}}$9 and $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$0 approaches $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$1 as $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$2 near the quadratic band touching, while the corresponding distance in non-touching bands approaches $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$3. In the same platform, robust boundary modes were simulated in finite kagome systems and shown to persist even under local perturbations, consistent with the real-space-topological interpretation (Karki et al., 2022).

The same loop-state logic has been generalized to planar systems with patterned hole defects. In holed two-dimensional systems, degeneracy can be stabilized by non-contractible loop excitations tied to hole defects, and the counting of flat-band excitations is expressed as

$\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$4

where compact localized excitations, robust boundary modes, and line modes all contribute. This construction is presented as a way to emulate topological degeneracy associated with nontrivial manifolds while remaining on a flat plane (Chen et al., 26 Mar 2025).

4. Quantum geometry and response to external fields

Singular flat bands are strongly constrained by quantum geometry. The review literature identifies the maximum quantum distance,

$\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$5

and its local maximum near the singularity as a bulk invariant representing the strength of the singularity. In this description, the singularity itself is topologically trivial, but the maximum quantum distance protects robust boundary modes and controls magnetic response (Rhim et al., 2020).

One of the clearest manifestations is anomalous Landau-level spreading. In a two-band continuum model for a singular flat band with quadratic band crossing, the anomalous Landau-level spreading $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$6 is governed by the maximal quantum distance $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$7, and the exact solution yields two branches of $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$8 corresponding to the two chiralities of the singular-flat-band wave function. In a diatomic kagome lattice hosting two singular flat bands mirrored by particle-hole symmetry, $\alpha_{\mathbf{k}} |\psi_{\mathbf{k}}\rangle$9 depends not only on $\alpha_{\mathbf{k}}$0 but also on the real-space diatomic distance $\alpha_{\mathbf{k}}$1; as $\alpha_{\mathbf{k}}$2 increases, $\alpha_{\mathbf{k}}$3 shrinks toward zero while $\alpha_{\mathbf{k}}$4 remains intact. The paper attributes this to magnetic-field-induced disruption of destructive interference in the compact localized states and derives the $\alpha_{\mathbf{k}}$5 dependence from the tuning of the non-Abelian orbital moment by real-space geometry (Long et al., 5 May 2025).

The electric-field response is similarly geometric. In a minimal two-band lattice model under a static uniform electric field, the Wannier-Stark spectrum of the singular flat band is, away from the band crossing point, captured by an intraband Berry phase, and the corresponding Wannier-Stark eigenstates are exponentially localized along the field direction, precluding dc transport. Near the band crossing point, however, the interband Berry connection drives Landau-Zener tunneling, bends the Wannier-Stark ladder, and delocalizes the singular-flat-band wave functions. This regime is governed solely by the maximal quantum distance $\alpha_{\mathbf{k}}$6 through two geometric phases $\alpha_{\mathbf{k}}$7, where $\alpha_{\mathbf{k}}$8 controls the tunneling rate and $\alpha_{\mathbf{k}}$9 acts as a generalized Berry phase (Long et al., 27 Jan 2026).

The three-dimensional extension exhibits additional structure. In the pyrochlore lattice, a pair of degenerate flat bands touches a dispersive band, and a three-orbital effective continuum model identifies a point-like topological singularity on the planar manifold defined by the degenerate flat-band eigenvectors. Under magnetic field, the orbital Zeeman effect lifts the flat-band degeneracy, reconstructs the low-energy spectrum into a Weyl-semimetal-like form near the singularity, and produces Landau-level spreading whose energy range is proportional to the quantum metric of each Zeeman-split band (Kawakami et al., 17 Jun 2025).

5. Interactions, superconductivity, and fractional topological phases

Because flat bands have a large density of states, singular flat bands are natural settings for interaction-driven phenomena. In topological media with nodal lines or Weyl points, bulk-surface and bulk-vortex correspondence can generate flat bands of fermion zero modes localized on surfaces or in vortex cores, and the resulting density of states is described as extremely singular. In that setting, the flat-band self-consistency equation for superconductivity becomes linear rather than exponentially suppressed, yielding a transition temperature that scales linearly with the interaction strength; the same singularity can enhance tendencies toward other symmetry-breaking orders (Heikkila et al., 2010).

For singular band-gap systems, the quantum geometry of the band touching enters directly into phase stiffness. A 2024 study of zero-energy flat bands in proximity to high-energy bands shows that when the singular band gap closes, the geometric and conventional contributions to the superfluid weight exhibit different crossover behaviors as functions of the superconducting gap. In the weak-coupling, gap-closing regime, the geometric contribution acquires a logarithmic enhancement, and tuning the singular band gap enhances the Berezinskii-Kosterlitz-Thouless transition temperature in two-dimensional superconductors (Jiang et al., 2024).

Fractional topological phases need not be restricted to isolated flat Chern bands. In the bipartite limit of the nearest-neighbor tight-binding model of twisted bilayer MoTe$\alpha_{\mathbf{k}}$0, a singular flat band supports fractional quantum anomalous Hall phases at $\alpha_{\mathbf{k}}$1 and $\alpha_{\mathbf{k}}$2 filling, demonstrated by density matrix renormalisation group calculations with all bands and exact diagonalization calculations with the two touching bands. In that model, gapping the band touching turns the singular flat band into a nearly flat Chern band, but suppresses the fractional quantum anomalous Hall effect because the gap opening introduces strong inhomogeneity to the quantum geometry (Yang et al., 2024).

The non-Chern setting can be pushed further. In a two-orbital honeycomb model, a singular gapless flat band at $\alpha_{\mathbf{k}}$3 supports a bosonic Laughlin state at half filling, stabilized by onsite interactions from the hard-core limit down to arbitrarily small strength, and also a bosonic Moore-Read state at $\alpha_{\mathbf{k}}$4 filling with a three-body hard-core constraint. In a distinct study of gapless flat bands with divergent quantum geometry, exact diagonalization and density matrix renormalization group calculations show $\alpha_{\mathbf{k}}$5 fractional quantum anomalous Hall phases that persist from the weak-interaction to the strong-interaction limit, with stability controlled not uniquely by singularity strength but by an occupation-weighted Berry flux shaped by an inhomogeneous carrier distribution (Lu et al., 16 Oct 2025, Yang et al., 17 Dec 2025). This suggests that exact flatness and adaptive many-body occupation can stabilize fractionalized phases even when band topology is ill-defined.

6. Model systems and experimental realizations

Singular flat bands have been realized or modeled in several photonic and acoustic platforms. In a photonic super-Kagome lattice with a nine-site unit cell, the upper two flat bands are singular and touch neighboring dispersive bands at the Brillouin-zone center, whereas the lower three degenerate flat bands are nonsingular and spectrally isolated. The classification is carried out both in momentum space, through singularities of the Bloch wave functions, and in real space, through the number of unit cells occupied by the compact localized state: the singular flat-band compact localized states occupy $\alpha_{\mathbf{k}}$6 unit cells, while the nonsingular ones are more compact (Song et al., 2023).

A frustrated Sierpinski fractal-like photonic lattice realizes both classes in a single structure: two isolated and degenerate nonsingular flat bands at $\alpha_{\mathbf{k}}$7 and one singular flat band at $\alpha_{\mathbf{k}}$8 due to band touching at $\alpha_{\mathbf{k}}$9. Diffractionless propagation of the nonsingular compact localized states was observed experimentally, and linear combinations of translated nonsingular compact localized states were shown to propagate without diffraction, consistent with completeness of the compact-localized-state basis in the nonsingular sector (Hanafi et al., 2021).

In tunable acoustic metamaterials based on thin-plate resonators, non-singular flat bands arise in triangular and honeycomb lattices by fine-tuning the ratio of global tension to bending stiffness, while a singular kagome flat band arises from the underlying lattice geometry and can be made degenerate with two additional flat bands by tuning the plate tension. The singularity is quantified through the Hilbert-Schmidt distance at the quadratic band touching, and robust boundary modes were simulated as an acoustic manifestation of real-space topology (Karki et al., 2022).

The modified Haldane-Dice model provides a tunable electronic example in which uniaxial strain and Haldane-type next-nearest-neighbor terms move and merge the band-touching points without removing singularity. When the central band is flat, its Hilbert-Schmidt quantum distance reaches the maximal value $\mathbf{k}_0$0 at all band-touching points, and explicit non-contractible loop states can be written on the real-space torus (Filusch et al., 2023).

7. Extensions, adjacent concepts, and unresolved structure

Recent work has sharpened the distinction between standard singular flat bands and related constructions with well-defined topology. In the conventional Lieb and kagome cases, the flat-band Bloch state vanishes at the touching point, the projector is discontinuous there, and topological invariants for the entire flat band are ill-defined. A 2026 theory proposes a second topological condition, namely linear dependence of loop states in different directions,

$\mathbf{k}_0$1

which removes the singularity at the band touching point while preserving non-contractible loop-state structure. The resulting “topological-topological” flat bands have well-defined Chern, $\mathbf{k}_0$2, and topological-crystalline invariants and flow, under small generic interactions, to correlated topological insulators with a dynamically generated symmetric mass term (Liu et al., 25 Mar 2026).

Another adjacent direction concerns the relation between singular or nearly singular flatness and van Hove physics. A 2024 review on high-order van Hove singularities emphasizes that nearly flat bands naturally harbor points of high local band flatness and power-law divergence in the density of states. In that framework, the local dispersion near the critical point is expanded as

$\mathbf{k}_0$3

and the corresponding density of states follows a power law

$\mathbf{k}_0$4

The review frames high-order van Hove points as a weakly dispersive counterpart of exactly flat-band physics and discusses their connection to interaction effects and engineered band structures (Classen et al., 2024).

These developments clarify two open structural issues. First, singular flat bands are not synonymous with topological flat bands in the sense of well-defined global invariants; their hallmark is the immovable Bloch-wave discontinuity and the concomitant incompleteness of compact localized states. Second, singularity can be either preserved, quantified, and exploited—as in anomalous Landau levels, robust boundary modes, or gapless fractional phases—or deliberately removed by additional constraints, producing new flat-band families with different topological content.