Localized State Energy Bands (LSEBs)

• LSEBs are energy or propagation-constant manifolds defined by eigenstates that are spatially localized, including flat bands, line and loop states, and interface-localized minibands.
• They are constructed using tight-binding generators, local symmetry partitioning, and semiclassical matching, which reveal insights into lattice topology and the effects of disorder.
• Experimental platforms such as photonic lattices and semiconductor heterostructures validate LSEBs, demonstrating their roles in localized transport, interference phenomena, and topological transitions.

Localized State Energy Bands (LSEBs) denote energy- or propagation-constant manifolds whose eigenstates are localized in real space. In periodic Hermitian lattices the canonical LSEB is a flat band spanned by compact localized states (CLS), but the term also covers singular flat bands that require line or loop states for completeness, boundary- and corner-localized spectra, interface-localized minibands, quasiperiodic real-space bands $E_n(r)$ defined on a compact reduced-coordinate manifold, and non-Hermitian localized-state spectra in the complex plane (Maimaiti et al., 2016, Xia et al., 2018, Chen et al., 7 Sep 2025, Hetényi et al., 1 Aug 2025).

1. Canonical meaning: flat bands as localized-state bands

In the most standard usage, an LSEB is a flat Bloch band. For a one-dimensional periodic tight-binding lattice with $\nu$ sites per unit cell, Bloch’s theorem gives a Bloch Hamiltonian $H(k)$ with bands $E_\mu(k)$, and a flat band is defined by $E_\mu(k)=FB$ independent of $k$. The associated eigenstates can be chosen as compact localized states occupying a finite number $U$ of adjacent unit cells. Translating such a CLS along the lattice produces an infinite set of eigenstates with the same energy $FB$, and for $U=1,2$ the translated CLS form the flat-band Hilbert subspace (Maimaiti et al., 2016).

This localized-state interpretation is explicit in photonic Lieb lattices. In the nearest-neighbor coupled-mode description, the Bloch Hamiltonian has three bands,

$$\Omega_\pm(\mathbf{k})=\pm 2\sqrt{\kappa_x^2\cos^2(k_x a)+\kappa_y^2\cos^2(k_y a)},\qquad \Omega_0(\mathbf{k})=0,$$

so the middle band is perfectly flat. The flatness implies zero group velocity and an infinite effective mass, and the corresponding real-space eigenmodes are compact localized states supported on a small set of $\nu$0 and $\nu$1 sites with destructive interference on the $\nu$2 sites (Mukherjee et al., 2014).

A broader usage, adopted explicitly or effectively in later work, treats LSEBs as energy-resolved manifolds of localized states beyond ordinary Bloch flat bands. That broader scope includes localized spectra induced by impurities, interfaces, quasiperiodicity, or topology. This suggests that the essential feature is not strict translational invariance but the existence of a spectrally identifiable set of localized eigenstates that can be organized in band-like form.

2. Compact localized states, incompleteness, and real-space topology

A common misconception is that a flat band is always fully described by CLS alone. The Lieb lattice provides the standard counterexample. With three sites per unit cell $\nu$3, nearest-neighbor couplings, and chiral bipartite structure, the Bloch spectrum is

$\nu$4

The minimal CLS has support on two $\nu$5 sites and two $\nu$6 sites surrounding a single $\nu$7 site and zero amplitude on all $\nu$8 sites, but when the flat band touches the dispersive bands the CLS set becomes linearly dependent on a torus-like geometry. The missing states are exact zero-energy line states,

$\nu$9

with all other amplitudes zero. These states form noncontractible loops and cannot be written as a finite linear combination of compact clusters (Xia et al., 2018).

The distinction is therefore between contractible localized states and noncontractible localized states. In the Lieb case, the full flat-band manifold is spanned by CLS together with two topologically distinct line states, one winding along $H(k)$0 and one along $H(k)$1. Completeness is boundary-dependent: with “flat” edges, line states cannot terminate consistently and CLS form a complete basis; with “bearded” edges, line states remain valid eigenmodes and must be added. The relevant topology is real-space topology of the graph and its boundary conditions, not momentum-space topology.

The same completeness issue appears in frustrated fractal-like lattices. In the Sierpinski photonic lattice, the flat band at $H(k)$2 is singular because its Bloch eigenvector has an immovable discontinuity at $H(k)$3, so truncated-triangle CLS are incomplete and noncontractible loop states are required. By contrast, the doubly degenerate flat bands at $H(k)$4 are nonsingular and can be spanned completely by two families of compact “reindeer” CLS. The paper’s terminology makes the criterion precise: nonsingular flat bands admit a complete CLS basis, whereas singular flat bands do not (Hanafi et al., 2021).

3. Generators and localized-basis constructions

LSEBs can be generated constructively from local constraints. In one dimension, the flat-band generator of Flach and collaborators classifies flat-band networks by hopping range $H(k)$5, number of bands $H(k)$6, and CLS class $H(k)$7. For the nontrivial case $H(k)$8, the Hamiltonian can be written in canonical form

$H(k)$9

with flat-band condition

$E_\mu(k)$0

The resulting flat-band energy is

$E_\mu(k)$1

This yields the complete two-parameter family of irreducible two-band, nearest-neighbor, $E_\mu(k)$2 flat-band Hamiltonians, including the standard sawtooth chain and the high-symmetry ST2 chain with equal hoppings (Maimaiti et al., 2016).

A complementary design route uses local symmetry partitioning. For discrete Hamiltonians with local site permutations, the Equitable Partition Theorem block-diagonalizes the Hamiltonian into a divisor matrix $E_\mu(k)$3 describing extended states and smaller blocks $E_\mu(k)$4 whose eigenvectors are automatically compact localized states. Periodic repetition of the locally symmetric motif converts those CLS into flat energy bands. The framework also extends to restricted noncommuting local symmetries through the nonequitable partition theorem, allowing symmetry-induced bound states in the continuum and tunable flat bands in one and two dimensions (Röntgen et al., 2017).

Semiclassical multiwell systems realize the same structure without lattice discreteness at the microscopic level. For a one-dimensional potential with $E_\mu(k)$5 identical parabolic wells, one may construct $E_\mu(k)$6 localized approximate eigenstates $E_\mu(k)$7, each matching the $E_\mu(k)$8-th harmonic-oscillator state in one well and WKB tails in the barriers. Diagonalizing the Hamiltonian in that localized basis yields

$E_\mu(k)$9

with $E_\mu(k)=FB$0 exponentially small in the barrier action. The same formula follows from direct matching of exact parabolic-cylinder solutions and WKB wavefunctions. In the large-$E_\mu(k)=FB$1 limit this reproduces the widths of the narrow Mathieu bands at leading order (Song, 2016).

4. Boundary-, corner-, and interface-localized bands

LSEBs are not restricted to bulk flat bands. In a two-dimensional extension of the SSH model with chiral symmetry, one obtains one-dimensional zero-mode bands localized at twin boundaries, antiphase boundaries, and open edges. The relevant topological invariant is a directional winding number,

$E_\mu(k)=FB$2

defined for each boundary orientation. A nonzero difference $E_\mu(k)=FB$3 across the boundary yields flat edge or interface bands pinned at zero energy. Their dispersion can be made strictly flat for well-separated boundaries and weakly dispersive for nearby boundaries, with the average group velocity controlled by the boundary spacing (Zhu et al., 2018).

Higher-order topology gives a discrete rather than continuous variant of the same idea. In the quadrupole topological insulator of Benalcazar–Bernevig–Hughes type, an edge with $E_\mu(k)=FB$4 corners supports $E_\mu(k)=FB$5 localized corner modes in the decoupled limit. Chiral symmetry enforces spectral symmetry about zero. If $E_\mu(k)=FB$6 is odd, one exact zero-energy localized state remains; if $E_\mu(k)=FB$7 is even, all localized levels split into $E_\mu(k)=FB$8 pairs and none stay at zero. The resulting corner-state manifold can be viewed as a finite LSEB whose structure is fixed jointly by bulk quadrupole topology and edge geometry (Takane, 2020).

Semiconductor heterostructures furnish an interface version of the same phenomenon. In ultrathin $E_\mu(k)=FB$9 superlattices, a 14-band $k$0 theory with an interface asymmetry Hamiltonian predicts localized energy levels induced by sub-nanometer interfacial broadening. These localized levels give an optical critical point $k$1 between $k$2 and $k$3 eV and extend absorption to lower energies. The shift of $k$4 with the interfacial width $k$5 provides a non-destructive optical probe of buried-interface broadening (Attiaoui et al., 2022).

5. Quasiperiodic, non-Hermitian, and topology-obstructed generalizations

For quasiperiodic lattices, the recent incommensurate-energy-band framework extends the notion of a band beyond translational symmetry. In the localized regime of the Aubry–André–Harper model, the relevant compact manifold is no longer the Brillouin zone but the real-space reduced coordinate $k$6, obtained by the spiral mapping $k$7. The localized-state spectrum is then described by LSEBs

$k$8

defined on a “real-space Brillouin zone.” The density of states takes the band-theoretic form

$k$9

For systems with mobility edges, such as the generalized Aubry–André–Harper model, the spectrum becomes hybrid: extended states are described by incommensurate energy bands $U$0, localized states by LSEBs $U$1, and the two sectors are separated by mobility edges (Chen et al., 7 Sep 2025).

Non-Hermitian impurity problems broaden the energy-space structure further. In a one-dimensional tight-binding ring with one special bond,

$U$2

a Hermitian impurity with $U$3 can generate two localized impurity states outside the tight-binding band once $U$4. With a non-Hermitian bond $U$5, $U$6, the localized pair can instead move onto the imaginary axis, and in a more general interpolation regime a macroscopic fraction of states localize around the impurity, producing a non-Hermitian skin effect. In the reported numerics, the fraction of states with $U$7 is approximately $U$8 for $U$9, identifying an extensive localized-state band in the complex plane (Hetényi et al., 1 Aug 2025).

A complementary limitation is provided by topological fine structure under disorder. A trivial band can fail to become fully localized under arbitrarily weak disorder because the localizer index $FB$0 changes within the band. In the cited example this produces two sets of extended states at two different energy intervals, carrying opposite Chern numbers. The implication is that LSEBs need not exhaust a band’s description: topology can force mobility edges and surviving extended states inside an otherwise localized spectrum (Liu et al., 2023).

6. Experimental platforms and dynamical consequences

Photonic lattices have provided the clearest direct realizations of LSEBs. In a femtosecond-laser-written photonic Lieb lattice, selective excitation of the flat band was achieved by exciting the two $FB$1 sites and two $FB$2 sites of a primitive cell with equal intensities and alternating phases, yielding a non-diffracting localized state in the flat middle band. The same platform showed that off-diagonal disorder preserves the exact flat band, whereas diagonal disorder destroys exact flatness (Mukherjee et al., 2014).

A second photonic development was the direct observation of line states in a finite Lieb lattice with tailored “bearded” boundaries. Continuous-wave laser writing in a photorefractive SBN crystal produced boundary terminations that support line states, and input beams shaped into out-of-phase Gaussian pearls remained intact after propagation through a $FB$3 mm crystal. The out-of-phase line self-extends to the full line state for bearded edges, whereas the same input fails to stabilize for flat edges. The same platform demonstrated shape-preserving transport of patterns extending to the boundary, including letters such as “P”, “R”, and “L” (Xia et al., 2018).

Localized-state dynamics need not be static. In the decorated photonic Lieb-5 lattice, the nearest-neighbor tight-binding spectrum contains two flat bands at $FB$4. Their compact localized states may be superposed to form oscillating compact localized states with beat frequency $FB$5, so the state remains compact while its internal intensity distribution oscillates along the propagation direction $FB$6. This establishes that multiple LSEBs can support coherent internal dynamics rather than only rigid non-diffracting transport (Hanafi et al., 2022).

Outside tight-binding photonic crystals, twisted bilayer systems realize superflat LSEBs centered on AA-stacked regions. In the reported theory, continuous lattice dislocation generates an effective macroscopic potential well around AA stacking, supporting intrinsic localized states at the lowest and highest energies. Because the inter-cell coupling of these AA-centered states is negligible, they form superflat bands $FB$7 and $FB$8 over a continuous range of small twist angles; the mechanism was also mimicked in twisted bilayer nanophotonic systems (Wang et al., 2022).

Finally, localization does not preclude transport by composite or embedded degrees of freedom. Arrays of bound states in the continuum can form a “crystal of BICs,” where continuum-mediated coupling generates a dispersive miniband embedded in radiating-wave continua (Longhi, 2021). In interacting flat-band lattices, flux-periodic numerics and exact intertwining symmetries show that the many-body spectrum is $FB$9-periodic, implying that only charge-$U=1,2$0 bound pairs propagate while single quasiparticles remain localized in the flat-band Wannier basis (Tovmasyan et al., 2018).

Localized State Energy Bands therefore comprise a family of spectral structures rather than a single mechanism. In the narrow sense they are flat bands generated by destructive interference and spanned by CLS; in the broader sense they include line and loop completions of singular flat bands, boundary and interface manifolds of localized states, quasiperiodic real-space bands, complex-energy non-Hermitian localized spectra, and interaction-generated transport sectors built atop localized single-particle manifolds. What unifies these realizations is that localization is organized spectrally: the localized states form a band, a miniband, or a clearly defined energy-resolved manifold whose structure can be derived from geometry, topology, symmetry, or effective tunneling.