Real space decay of flat band projectors from compact localized states

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.

Figures (2)

• Figure 1: The generalized 2D checkerboard lattice. Note that the black lines represent hoppings with tunable values. (a): The lattice structure. Black circles represent a CLS [See Eq. \ref{['eq:square_cls']}]. For the values of hoppings and CLS amplitudes, see SM supp. (b): The band structure. The two bands are $E_{\mathrm{FB}} = 0$, and $E_{\mathrm{DB}}(k) = \alpha^2(k)$. Here, $A = 0.5$. $\Delta$ is the band gap. (c) Algebraic part of the projector $P(x)e^{x/\xi}$ shown in log-log plot. The red and green dashed lines guide the eye for $x^{-1/2}$ and $x^{-2}$ respectively. The parameter $A=0.5$ places the FB model half way between orthogonal ($A=0$) and singular ($A=1$) limits. (d) Same as (c) but $A=0.99$. This places the FB model in close proximity to the singular limit. The vertical black dashed line indicates the value of the localization length $\xi_{\hat{x}} = 70$ (e): ratio $f(x) = P(1,1;x+1)/P(1,1;x)$ vs. $1/x$. Here $A=0.9$. The straight black dashed line is obtained from a linear fit of the smallest $1/x$ data (see inset). Its intercept value with the y-axis is $0.86$ and results in a numerical estimate of $\xi=6.7$ in excellent agreement with our analytics. The slope of the dashed line is $-0.54$ (after dividing by y-intercept) in excellent agreement with the analytical prediction $-{\rm e}^{-1/\xi}/2$ (the value gets close to -0.5 as larger system size increases). The departure of the data from the dashed line are observed precisely at $1/x=1/\xi$ as predicted by analytics and indicated by the vertical dotted line. (f): Localization length $\xi_{\hat{x}}$ versus $A$. Symbols - numerical results from the above intercept fitting. Dashed line - analytics.
• Figure 2: A generalized anisotropic Lieb lattice which hosts an anisotropic singular flat band. (a) Lattice, the black lines are hoppings with strength 1, and the red lines are hoppings with strength 2. The CLS corresponds to the black colored circles [See Eq. \ref{['eq:nongeneric_bcls']} and SM supp]. (b) Band structure of the lattice (a). (c) Decay of the projector $|P(1, 1; \mathbf{r})|$ vs. $\mathbf{r}$ in log-log scale. Red: $\mathbf{x}$ direction, linear fitting shows exponent $-3/2$, Blue: $\mathbf{y}$ direction, linear fitting shows exponent $-3$.