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arXiv:2510.17258 · Physical review. B./Physical review. B

Real-space decay of flat-band projectors from compact localized states

Yeongjun Kim, Sergej Flach A, Alexei Andreanov

Published Apr 29, 2026• 2 citations
arXivHTMLPDF

Abstract

Flatbands (FB) with compact localized eigenstates (CLS) fall into three main categories, controlled by the algebraic properties of the CLS set: orthogonal, linearly independent, linearly dependent (singular). A CLS parametrization allows us to continuously tune a linearly independent FB into a limiting orthogonal or a linearly dependent (singular) one. We derive the asymptotic real space decay of the flat band projectors for each category. The linearly independent FB is characterized by an exponentially decaying projector and a corresponding localization length $ξ$, all dressed by an algebraic prefactor. In the orthogonal limit, the localization length is $ξ=0$, and the projector is compact. The singular FB limit corresponds to $ξ\rightarrow \infty$ with an emerging power law decay of the projector. We obtain analytical estimates for the localization length and the algebraic power law exponents depending on the dimension of the lattice and the number of bands involved. Numerical results are in excellent agreement with the analytics. Our results are of relevance for the understanding of the details of the FB quantum metric discussed in the context of FB superconductivity, the impact of disorder, and the response to local driving.

DOI
10.1103/x5rg-lfkb

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Flatbands (FB) with compact localized eigenstates (CLS) fall into three main categories, controlled by the algebraic properties of the CLS set: orthogonal, linearly independent, linearly dependent (singular). A CLS parametrization allows us to continuously tune a linearly independent FB into a limiting orthogonal or a linearly dependent (singular) one. We derive the asymptotic real space decay of the flat band projectors for each category. The linearly independent FB is characterized by an exponentially decaying projector and a corresponding localization length ξξξ, all dressed by an algebraic prefactor. In the orthogonal limit, the localization length is ξ=0ξ=0ξ=0, and the projector is compact. The singular FB limit corresponds to ξ→∞ξ\rightarrow \inftyξ→∞ with an emerging power law decay of the projector. We obtain analytical estimates for the localization length and the algebraic power law exponents depending on the dimension of the lattice and the number of bands involved. Numerical results are in excellent agreement with the analytics. Our results are of relevance for the understanding of the details of the FB quantum metric discussed in the context of FB superconductivity, the impact of disorder, and the response to local driving.
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In flat-band systems with quenched kinetic energy, most of the conventional length scales related to the band dispersion become ineffectual. Although a few geometric length scales, such as the quantum metric length, can still be defined, because of their embedding dependence, i.e., the dependence on the choice of orbital positions used to construct the tight-binding model, they cannot serve as a universal length scale of the flat-band systems. Here, we introduce an embedding independent length scale ξflatξ_\text{flat}ξflat​ of a flat band that is defined as the localization length of an in-gap state proximate to the flat band. Because ξflatξ_\text{flat}ξflat​ is derived from the intrinsic localization of compact localized states, it is solely determined by the Hamiltonian and provides a robust foundation for embedding independent observables. We show analytically that the superconducting coherence length in a flat-band superconductor is given by ξflatξ_\text{flat}ξflat​ in the weak-coupling limit, thereby identifying ξflatξ_\text{flat}ξflat​ as the relevant length scale for many-body phenomena. Numerical simulations on various lattice models confirm all theoretical predictions, including the correspondence between ξflatξ_\text{flat}ξflat​ and the superconducting coherence length. Our results highlight ξflatξ_\text{flat}ξflat​ as a universal length scale for flat bands and open a pathway to embedding independent characterization of interacting flat-band materials.
In recent years, there has been a growing interest in flatband systems which exhibit macroscopic degeneracies. These systems offer a valuable mathematical framework for the extreme sensitivity to perturbations and interactions. This sensitivity unveils a wide variety of exotic and unconventional physical phenomena. Moreover, the progress in their experimental realization contributes to the expanding landscape of exploration in this field. This thesis aims to summarize all the works throughout the Ph.D. program. Firstly, an in-depth exploration was conducted on the impact of weak quasiperiodic perturbations on one-dimensional two-band all-bands-flat lattices. These tight-binding Hamiltonian are diagonalized through a sequence of local unitary transformations. By adjusting the quasiperiodic potential parameters, the key achievement involves finding a critical-to-insulator transition and identifying fractality edges in the flatband systems with quasiperiodic perturbations. Next, the investigation delved into the effects of on-site interactions among hard-core bosons in one- and two-dimensional cross-stitch lattices. One key finding is that groundstate energy primarily depends on compact localized states. Moreover, the presence of barriers of compact localized states trap bosons, leading to the emergence of non-ergodic excitation and Hilbert space fragmentation. Lastly, a compact localized eigenstate of the one-dimensional diamond chain using an electric circuit was successfully generated via local (linear and non-linear) driving. This achievement opens up a versatile circuit platform for the generation of flatbands and holds promise for potential applications in the field of quantum information. I hope these collective efforts have expanded the frontiers of the field and made a meaningful contribution to the scientific community.
The spatial confinement of defect-induced bound states is conventionally governed by the effective mass in dispersive bands. More recently, Compact Localized States (CLSs) arising from exact destructive interference have been utilized to achieve confinement in flat bands. However, CLSs rely on pristine lattice symmetries and fine-tuned defect profiles. The introduction of a generic local impurity inevitably breaks these strict phase-matching conditions, resulting in extensive bound states whose fundamental length scale has remained an open question. Here, we establish a third regime of confinement: the quantum metric bound state. We provide a rigorous mathematical proof demonstrating that in the absence of kinetic energy and CLS protection, the exponential decay length of these states is lower-bounded by the quantum metric of the unperturbed flat band. We demonstrate the tightness of this geometric limit by constructing a family of highly tunable flat-band generators, and we verify its universality across diverse realistic architectures. Ultimately, this classification establishes the independently measurable quantum metric as a predictive design principle for engineering confined modes in synthetic wave platforms.
Abstract Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1 d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.
The possibility of an unconventional form of high temperature superconductivity in flat band (FB) material does not cease to challenge our understanding of the physics in correlated systems. Here, we calculate the normal and anomalous one-particle correlation functions in various one and two dimensional FB systems and systematically extract the characteristic lengthscales. When the Fermi energy is located in the FB, it is found that the coherence length ( \xi ξ ) is of the order of the lattice spacing and weakly sensitive to the strength of the electron-electron interaction. Recently, it has been argued that in FB compounds \xi ξ could be decomposed into a conventional part of BCS type ( \xi_{BCS} ξ B C S ) and a geometric contribution which characterises the FB eigenstates, the quantum metric ( \langle g \rangle ⟨ g ⟩ ). However, by calculating the coherence length in two possible ways, our calculations show that \xi ≠ \sqrt{\langle g \rangle.} ξ ≠ ⟨ g ⟩ . This may suggest that the link between QM and coherence length is more complex, and leaves us with the open question: what is the appropriate definition of the coherence length in flat-band systems?