Compact Localized States (CLSs) in Lattice Physics

• CLSs are eigenstates with exact finite support arising from destructive interference in lattice Hamiltonians.
• In periodic systems, translated CLSs form flat bands with macroscopic degeneracy and are characterized by their minimal unit-cell support (class U).
• CLSs are realized across diverse platforms—from photonic lattices to many-body systems—leading to novel applications in flat band phenomena.

Searching arXiv for recent and foundational papers on compact localized states to ground the article in published work. Compact localized states (CLSs) are eigenstates of lattice or graph-based Hamiltonians whose amplitudes are strictly nonzero only on a finite set of sites and exactly zero elsewhere. In the tight-binding language, a representative form is

$$|\psi_{\mathrm{CLS}}\rangle=\sum_{n\in\mathcal S} c_n |n\rangle, \qquad c_n=0 \text{ for } n\notin\mathcal S,$$

with $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$. Their defining physical mechanism is destructive interference at the boundary of the support region, which suppresses leakage into the surrounding lattice. In periodic systems, translated copies of a CLS generate a macroscopically degenerate eigenspace and thus a flat band $E_\alpha(k)=E_0$, while in nonperiodic settings CLSs remain exact compact eigenmodes without requiring translational invariance (Röntgen et al., 2017).

1. Definition, compactness, and relation to flat bands

A CLS is distinguished from exponentially localized modes by its exact finite support. The compactness is structural rather than disorder-driven: the wavefunction vanishes identically outside a small subset, and because it is an eigenstate, this profile is preserved in time. In the flat-band literature this is the standard real-space manifestation of a dispersionless Bloch band, since translated CLSs remain eigenstates with the same eigenvalue and can be recombined into Bloch eigenvectors with momentum-independent energy (Maimaiti et al., 2016).

In one-dimensional translationally invariant lattices, a useful refinement is the CLS class $U$, defined as the minimal number of adjacent unit cells occupied by the state. For nearest-neighbor two-band chains, this classification is not merely descriptive: it organizes flat-band generators and constrains what kinds of compact modes are possible. In the canonical examples treated in one dimension, $U=1$ corresponds to orthogonal one-cell CLSs, while $U=2$ captures irreducible two-cell structures such as those appearing in sawtooth-type models (Maimaiti et al., 2016).

The same real-space logic extends beyond simple single-orbital Bravais lattices. In higher-dimensional tight-binding constructions, a CLS is still characterized by a finite cluster $\mathcal C$ together with boundary conditions

$\sum_{i\in\mathcal C} t_{ji}\psi_i=0$

on every site $j$ outside $\mathcal C$ that is connected to the cluster. This condition encodes the destructive interference that underlies compactness and is the starting point for systematic CLS design in single-orbital, multi-orbital, quasicrystalline, and disordered settings (Chen et al., 2022).

2. Structural origin: destructive interference, local symmetry, and graph partitioning

The most direct explanation of CLS formation is boundary decoupling by destructive interference. In one-dimensional flat-band generators this appears as zero-leakage constraints such as

$$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$0

which ensure that amplitudes at the edge of the compact support do not feed neighboring unit cells. In canonical sawtooth examples, the relative amplitudes $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$1 or $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$2 implement precisely this cancellation, producing exact $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$3 compact eigenstates (Maimaiti et al., 2016).

A more structural formulation identifies CLSs as consequences of local permutation symmetries of discrete Hamiltonians. For tight-binding-type matrices $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$4, a local symmetry $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$5 with permutation matrix $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$6 yields

$$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$7

in the commutative case. The resulting equitable partition theorem decomposes the reordered Hamiltonian into

$$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$8

so that the spectrum splits into extended states governed by $$H|\psi_{\mathrm{CLS}}\rangle=E|\psi_{\mathrm{CLS}}\rangle$$9 and compact localized states governed by the smaller matrices $E_\alpha(k)=E_0$0. The corresponding CLS eigenvectors have zero amplitude on the fixed sites and live entirely inside the locally symmetric subsystem, making the destructive-interference mechanism explicit in block form (Röntgen et al., 2017).

This symmetry-based picture survives beyond Hermitian short-range lattices. In open scattering media, the dyadic Green’s matrix of dipolar nanoparticle arrays can be partitioned by the equitable partition theorem when the geometry has local mirror symmetry. The resulting compact localized scattering resonances are exact quasi-modes with support on a finite subset of dipoles and zero amplitude elsewhere, despite the presence of non-Hermitian, retarded, long-range couplings. A plausible implication is that CLSs are better regarded as a symmetry-enforced spectral phenomenon than as a peculiarity of nearest-neighbor tight-binding models (Sgrignuoli et al., 2018).

3. Classification: CLS class, singularity, and basis completeness

Two complementary classification schemes recur in the literature. The first is the real-space class $E_\alpha(k)=E_0$1, which measures the minimal unit-cell support of a CLS. The second is the singular versus nonsingular distinction for flat bands, which concerns whether Bloch eigenvectors are globally well-defined and whether translated CLSs span the entire flat-band subspace.

In nonsingular flat bands, Bloch eigenvectors can be chosen continuous across the Brillouin zone, and a complete compact basis exists. In the one-dimensional mechanical triangular lattice, for example, the isolated flat band has a well-defined normalized eigenvector across the Brillouin zone and admits an exact CLS occupying two unit cells and four sites. In that setting, the flat band is explicitly described as nonsingular, with the compact basis exhausting the flat-band subspace (Riva et al., 2024).

By contrast, singular flat bands are typically tied to band touching and to real-space obstructions. In the photonic super-Kagome lattice, the upper two flat bands are singular because they touch neighboring dispersive bands at the Brillouin-zone center, whereas the lower three degenerate flat bands are nonsingular and spectrally isolated. The associated CLSs reflect this distinction: the nonsingular sector supports compact states with small unit-cell occupancy, while the singular sector is associated with larger plaquette-supported CLSs and the need for a broader flat-band basis structure (Song et al., 2023).

The same pattern appears in fractal-like photonic lattices derived from first-generation Sierpinski geometry. There, a singular nondegenerate flat band coexists with two nonsingular degenerate flat bands. The singular band supports a representative CLS but not a complete compact spanning set, whereas the degenerate nonsingular flat-band pair can be reorganized into an analytic Bloch basis and correspondingly compact real-space states (Xie et al., 2021).

This incompleteness issue is also explicit in recent work on flat-band Bose-Einstein condensation. In that context, CLSs can always be constructed for finite-range flat bands, but singular flat bands require the inclusion of non-contractible loop states to complete the flat-band Hilbert space. This suggests that the distinction between singular and nonsingular flat bands is not merely spectral; it governs whether CLSs alone provide an adequate basis for interacting and condensate problems (Huhtinen, 10 Mar 2026).

4. Realizations across lattice, photonic, mechanical, and graph settings

CLSs occur in a wide range of platforms. In finite, disorder-free hypercube graphs built via Cartan products, even-dimensional zero-potential hypercubes possess a macroscopically degenerate zero-energy sector. Appropriate linear combinations of the resulting extended $E_\alpha(k)=E_0$2 eigenvectors produce exact zero-energy CLSs with support on only a few vertices, despite the absence of conventional translational invariance. The same framework also supports non-compact localized states under structured onsite potentials, clarifying the distinction between compact and Anderson-like localization in ordered high-dimensional graphs (Arkhipov et al., 2024).

In spinor flat-band systems with spin-orbit coupling, compactness can coexist with internal multicomponent structure and nonlinearity. For the diamond chain with Rashba-type spin-orbit coupling, the flat bands remain at $E_\alpha(k)=E_0$3, while the linear CLSs become class-$E_\alpha(k)=E_0$4 spinor states occupying two unit cells. Under the interaction constraint $E_\alpha(k)=E_0$5, these same profiles persist as exact nonlinear CLSs, with frequency shift

$E_\alpha(k)=E_0$6

providing a rare example where compactness survives onsite cubic nonlinearity in closed analytic form (Gligorić et al., 2016).

In elastic systems, CLSs appear as exact finite-support eigenmodes of spring-mass and MEMS lattices. In the triangular mechanical chain, a flat dispersion band $E_\alpha(k)=E_0$7 yields a compact mode with amplitudes on only four sites across two neighboring cells. Tailored excitation of the CLS profile excites only the flat-band subspace, leading to non-propagating dynamics and a single sharp resonance, whereas point excitation spreads across dispersive bands (Riva et al., 2024).

Photonic experiments provide direct real-space visualization. In super-Kagome and edge-centered triangular lattices, laser-written waveguide arrays support multiple CLS families with distinct geometrical supports. Alternating-phase inputs matched to the compact eigenmodes propagate without diffraction, while equal-phase excitations on the same sites diffract strongly. These experiments establish that precise phase structure is not an auxiliary detail but the operational content of destructive interference (Song et al., 2023).

5. Interactions, many-body generalizations, and condensate stability

The many-body extension of the CLS concept proceeds by mapping the interacting Hilbert space to a configuration-space network whose nodes are Fock states and whose edges are Hamiltonian matrix elements. A many-body CLS is then an eigenstate whose amplitudes are nonzero only on a finite connected subgraph of that network. This construction allows interacting Hamiltonians to be designed so that compact many-particle states are preserved, or even expanded, by projecting interactions onto the many-body extended-state subspace or by applying origami rules in the network representation (Santos et al., 2020).

This framework also underlies exact many-body scar constructions. In the saw-tooth flat-band model, orthogonal single-particle CLSs created by

$E_\alpha(k)=E_0$8

generate a degenerate manifold of low-entanglement many-body states at low filling. A weak linear potential lifts the degeneracy, and the perturbed descendants of these CLS many-body states appear as multiple quantum many-body scars embedded in an otherwise non-integrable spectrum, with slow thermalization and anomalously low entanglement entropy (Kuno et al., 2021).

A different interacting perspective emerges in one-dimensional fermionic systems with nilpotent hopping matrices. There, the operators

$E_\alpha(k)=E_0$9

define CLS fermions supported on two neighboring sites, and the Hamiltonian becomes ultra-local in the $U$0 basis. This basis is directly tied to lattice supertranslations and Carrollian symmetry, and it allows interacting flat-band models to be written in terms of strictly local CLS densities, with phase structure and criticality encoded in occupation patterns of compact modes (Ara et al., 2024).

For flat-band Bose-Einstein condensation, CLSs provide a real-space basis in which the mean-field minimization problem becomes Euclidean geometry. Uniform-density condensates correspond to geometric frameworks built from CLS coefficients; triangulated frameworks with nonzero area are favorable for stable condensation, whereas square frameworks imply additional zero modes in Bogoliubov theory and rule out condensation in a single mode. When the condensate is restricted to Bloch states, this criterion becomes related to a necessary condition for non-vanishing quantum distance (Huhtinen, 10 Mar 2026).

6. Impurities, disorder, BICs, and broader significance

CLSs are often described as delicate because they rely on exact interference, but the literature shows a more differentiated picture. In the diamond chain with magnetic flux, overlapping non-orthogonal CLSs span a gapped midspectrum flat band. Projecting a local impurity onto this non-orthogonal flat-band basis yields an effective two-state problem in the dual CLS subspace. For equal impurities on the top and bottom sites, the impurity states experience an averaged disorder determined by their spatial extension, leading to enhanced robustness against diagonal disorder. For a single impurity, one state remains pinned at zero energy while the effective two-band problem acquires a half-integer winding number linked to a single in-gap edge state under open boundary conditions (Marques et al., 2024).

Disorder can also generate, rather than destroy, compact localization. In the random quantum comb, configurational disorder in tooth lengths produces an effective one-dimensional backbone problem with resonant energies at which extensive families of CLSs appear. These states have exactly zero backbone localization length and can be assembled into many-body scar states that survive nearest-neighbor density interactions along the backbone (Hart et al., 2020).

A related but conceptually distinct application is the construction of tunable bound states in the continuum. In locally symmetric tight-binding defects embedded in an otherwise dispersive host, the equitable partition theorem isolates a compact mode on the defect while the divisor matrix can be tuned to coincide with the unperturbed host Hamiltonian. When the CLS energy lies inside the host continuum, the state becomes a bound state in the continuum without modifying the surrounding band structure (Röntgen et al., 2017).

Taken together, these developments show that CLSs are not confined to one narrow corner of flat-band physics. They are exact finite-support eigenstates that organize the spectra of periodic lattices, disorder-free graphs, open scattering media, mechanical metamaterials, and interacting many-body systems. A common misconception is that they are merely localized wave packets in flat bands; more precisely, they are structural eigenmodes whose existence, completeness, and stability are controlled by interference, local symmetry, overlap geometry, and, in singular cases, by the necessity of supplementing the compact basis with additional extended states (Röntgen et al., 2017).