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Embedding independent length scale of flat bands

Seokju Lee, Seung Hun Lee, Bohm-Jung Yang

TL;DR

The paper defines an embedding independent length scale $ξ_{flat}$ for flat-band systems by analyzing the localization of an in-gap state induced by a local perturbation, showing $ξ_{flat}$ is fixed by CLS overlaps and independent of orbital embedding. It proves that in the weak-coupling flat-band superconductor limit the superconducting coherence length $ξ_{coh}$ equals $ξ_{flat}$, linking a many-body observable to a single-particle localization property. The work connects $ξ_{flat}$ to the embedding-dependent quantum metric length $ξ_{QM}$ through explicit bounds, and demonstrates the theory with one-dimensional flat-band models including the Stub lattice, Sawtooth, and Lieb lattices, with extensions to nearly flat bands. These results provide a universal, embedding-independent framework for characterizing interacting flat-band materials and offer concrete guidance for interpreting superconducting and localization phenomena in such systems.

Abstract

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 $ξ_\text{flat}$ of a flat band that is defined as the localization length of an in-gap state proximate to the flat band. Because $ξ_\text{flat}$ 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 $ξ_\text{flat}$ in the weak-coupling limit, thereby identifying $ξ_\text{flat}$ as the relevant length scale for many-body phenomena. Numerical simulations on various lattice models confirm all theoretical predictions, including the correspondence between $ξ_\text{flat}$ and the superconducting coherence length. Our results highlight $ξ_\text{flat}$ as a universal length scale for flat bands and open a pathway to embedding independent characterization of interacting flat-band materials.

Embedding independent length scale of flat bands

TL;DR

The paper defines an embedding independent length scale for flat-band systems by analyzing the localization of an in-gap state induced by a local perturbation, showing is fixed by CLS overlaps and independent of orbital embedding. It proves that in the weak-coupling flat-band superconductor limit the superconducting coherence length equals , linking a many-body observable to a single-particle localization property. The work connects to the embedding-dependent quantum metric length through explicit bounds, and demonstrates the theory with one-dimensional flat-band models including the Stub lattice, Sawtooth, and Lieb lattices, with extensions to nearly flat bands. These results provide a universal, embedding-independent framework for characterizing interacting flat-band materials and offer concrete guidance for interpreting superconducting and localization phenomena in such systems.

Abstract

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 of a flat band that is defined as the localization length of an in-gap state proximate to the flat band. Because 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 in the weak-coupling limit, thereby identifying as the relevant length scale for many-body phenomena. Numerical simulations on various lattice models confirm all theoretical predictions, including the correspondence between and the superconducting coherence length. Our results highlight as a universal length scale for flat bands and open a pathway to embedding independent characterization of interacting flat-band materials.

Paper Structure

This paper contains 16 sections, 1 theorem, 157 equations, 9 figures.

Table of Contents

  1. Embedding dependence of quantum metric length
  2. Localization length of in-gap state
  3. Localization length near a dispersive band
  4. Localization length near a flat band
  5. Decay of in-gap states induced by local perturbation
  6. Paley-Wiener theorem
  7. $\xi_\text{flat}$ of several 1D flat-band systems
  8. Proof of $\xi_{\mathrm{coh}}=\xi_\text{flat}$ in flat-band superconductors
  9. Decay length $\xi_u$ and $\xi_F$
  10. Numerical calculation of $\xi_\text{coh}$
  11. Relation between $\xi_\text{flat}$ and $\xi_\text{QM}$
  12. Nonzero $\lambda_t$ for $t\geq2$
  13. Application to nearly flat bands
  14. Effective localization length for a nearly flat band
  15. Coherence length in a nearly flat band
  16. ...and 1 more sections

Key Result

Theorem 1

Let $f(z)$ be $2\pi$-periodic. Then $f(z)$ admits a holomorphic extension to the strip $|\text{Im}(z)|<\gamma$ iff its Fourier coefficients satisfy Moreover, the maximal width of the analytic strip is determined by the exponential decay rate: if $f$ does not extend beyond $|\Im z|<\gamma$, then one has so that decay faster than $e^{-\gamma|n|}$ is impossible.

Figures (9)

  • Figure 1: (a) Wave function profile of an exponentially localized in-gap state with localization length $\xi$. (b) A gapped band structure with an in-gap state. As the in-gap state energy (dashed red) approaches a dispersive bulk band (black), $\xi$ diverges. (c) When the in-gap state approaches a flat band, $\xi$ remains finite.
  • Figure 2: (a) Stub lattice model with three orbitals ($A,B,C$) per unit cell and hopping amplitudes $J$ and $Jd$. (b) Real-space profile of the in-gap state induced by a point impurity. The localization length extracted from an exponential fit is shown together with the theoretical prediction. The inset shows the band structure. The data are obtained for $J=1$ and $d=0.5$, in which applying a local potential $U=0.01$ to a single $C$ orbital generates an in-gap state at $E=0.0024$.
  • Figure 3: (a) Anomalous correlation function $|K_A(x)|$ on the $A$ sublattice in the superconducting Stub lattice for $J=10$ and $U=0.1$. (b) $\xi_\text{coh}$ (markers) and $\xi_\text{flat}$ (red line) as a function of $d$.
  • Figure S1: (a) Isolated dispersive band (blue) and real-energy arc of in-gap solutions in the complex-$k$ plane (red). As the in-gap energy approaches the band minimum, $\mathrm{Im}\,\mathbf{k}\to 0$ and $\xi\to\infty$. (b) Flat-band plane (cyan) and crossings (yellow) with real-energy arcs from other bands. Near a flat band, the controlling decay is set by the smallest $|\mathrm{Im}\,\mathbf{k}|$ among such crossings, leading to a finite $\xi$.
  • Figure S2: (a) Numerical result for the sawtooth lattice. The localization length of an in-gap state near the flat band is extracted by fitting. (b) Numerical result for the 1D Lieb lattice. (c) Numerical result for the Creutz ladder.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem : Paley-Wiener