Equivalence classes of Fibonacci lattices and their similarity properties

We investigate, theoretically and experimentally, the properties of Fibonacci lattices with arbitrary spacings. Different from periodic structures, the reciprocal lattice and the dynamical properties of Fibonacci lattices depend strongly on the lengths of their lattice parameters, even if the sequence of long and short segment, the Fibonacci string, is the same. In this work we show that by exploiting a self-similarity property of Fibonacci strings under a suitable composition rule, it is possible to define equivalence classes of Fibonacci lattices. We show that the diffraction patterns generated by Fibonacci lattices belonging to the same equivalence class can be rescaled to a common pattern of strong diffraction peaks thus giving to this classification a precise meaning. Furthermore we show that, through the gap labeling theorem, gaps in the energy spectra of Fibonacci crystals belonging to the same class can be labeled by the same momenta (up to a proper rescaling) and that the larger gaps correspond to the strong peaks of the diffraction spectra. This observation makes the definition of equivalence classes meaningful also for the spectral and therefore dynamical and thermodynamical properties of quasicrystals. Our results apply to the more general class of quasiperiodic lattices for which similarity under a suitable deflation rule is in order.

Figure 1

The composition rules ($LS\to L^{\prime }$, $L\to S^{\prime }$) on a semi-infinite Fibonacci lattice reproduce another Fibonacci lattice with different lattice parameters.

Figure 2

Inverse of KL divergence $1/D({\eta }_0^a,\eta ,\nu )$ comparing the diffraction spectra of the generator of a given class $x_n^{{\eta }_0}$ with another Fibonacci lattice, for two choices of ${\eta }_0$: (a) ${\eta }_0^a=6/11$ and (b) ${\eta }_0^b=1/6$. Here $\mathrm{\Delta }{\eta }^a=\eta -{\eta }_1^a$, $\mathrm{\Delta }{\nu }^a=\nu -{\eta }_1^a$ and $\mathrm{\Delta }{\eta }^b=\eta -{\eta }_1^b$, $\mathrm{\Delta }{\nu }^b=\nu -{\eta }_1^b{\eta }_2^b{\eta }_3^b$. The maxima [minima of $D({\eta }_{\alpha },{\eta }_{\beta },\nu )$] are obtained at $\mathrm{\Delta }{\eta }^{a,b}=0$ and $\mathrm{\Delta }{\nu }^{a,b}=0$, indicating that the two spectra with the highest degree of similarity correspond to lattices ${\eta }_1^a{x}_n^{{\eta }_1^a}=\mathcal{C}\left(x_n^{{\eta }_0^a}\right)$ and ${\eta }_1^b{\eta }_2^b{\eta }_3^b{x}_n^{{\eta }_3^b}={\mathcal{C}}^{\left(3\right)}\left(x_n^{{\eta }_0^b}\right)$ respectively, i.e., the first and the third elements of the respective equivalence classes. (c) Direct comparison of the two diffraction spectra for two FLs $x_n^{{\eta }_0^a}$ and ${\eta }_1^a{x}_n^{{\eta }_1^a}$. The most prominent peaks of the diffraction pattern $I(q,{\eta }_0^a)$ correspond to those of the (rescaled) spectrum $I({\eta }_1^{a}q,{\eta }_1^a)$.

Figure 3

(a) Comparison between the theoretical diffraction pattern for the FL $x_n^{{\eta }_0^a}$ (${\eta }_0^a=0.5454$, solid blue curve) and the experimental one produced by a FL $\mathcal{C}\left(x_n^{{\eta }_0^a}\right)$ (solid red curve, $N=300$ lines and $L=23\mu \mathrm{m}$ and $S=17\mu \mathrm{m}$, ${\eta }_1^a=2.8\overline{3}$). The theoretical spectrum has been rescaled in accordance with Eq. (4) and corrected for the structure factor contribution. (b) The inverse of KL divergence $\left[D^{-1}({\eta }_1^a,\eta ,\nu )\right]$ between the (normalized over the interval) experimental diffraction pattern and the theoretical diffraction patterns for different generators and different scaling. Here $\mathrm{\Delta }{\eta }^a=\eta -{\eta }_0^a$ and $\mathrm{\Delta }\nu =\nu -1/{\eta }_0^a$. The maximum (minima of $D$) is at $(\eta ,\nu )=({\eta }_0^a,1/{\eta }_0^a)$.

Figure 4

Plot of the energy level spacings of the Hamiltonian in Eq. (7) for a potential $V\left(x\right)$ having minima at the points of FLs $x_n^{\eta }$ with (blue dots) $\eta ={\eta }_0^a$ and (red crosses) ${\eta }_1^a$. Below we compare the gaps with the brightest peaks of $I(q/{\eta }_1^a,{\eta }_1^a)$.

Figure 5

Construction of a generalized Fibonacci lattice from a 2D periodic lattice by means of the cut and project method. Blue dots are the projection of points of the 2D lattice whose Vonröi cells are cut by the line $l_{\tau }$.

Figure 6

(a) Values of $r=|{\mathbf{e}}_1|/|{\mathbf{e}}_2|$ as a function of $\eta$. The red dot corresponds to the point $(\eta ,r)=(\tau ,1)$ for which the canonical Fibonacci lattice is obtained. (b) Values of the angle $\alpha$ between ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ as a function of $\eta$. The red dot corresponds to the point $(\eta ,\alpha )=(\tau ,\pi /2)$ for which the canonical Fibonacci lattice is obtained.

Figure 7

Comparison between the values of $I\left(q_{\parallel }\right)$ using the cut and project method and the direct evaluation in Eq. (B1) for a lattice of $N=300$ points and $\eta =17/6$. Blue dots are points corresponding to the value in Eq. (B14) whereas red crosses are given by Eq. (B1).

Figure 8

Plot of (blue dots) ${\tau }^{-1}[1-I(q,{\eta }_0)]$ and (red crosses) $[I({\eta }_{1}q,{\eta }_1)-I(q,{\eta }_0)]$ for those $q$ for which $I(q,{\eta }_0)>0.5$ showing that the intensities of the brightest peaks of the diffraction pattern of a FL $x_n^{{\eta }_1}$ generated from a FL $x_n^{{\eta }_0}$ by means of the composition rule are more intense than those of the original lattice by a factor of $\tau$.

Figure 9

Experimental diffraction pattern of the periodic grating with spacing $L=23\mu \mathrm{m}$ (blue, solid curve) and fit of the satellite peak intensities using function (E2) (red, dashed curve).

Figure 10

Experimental diffraction pattern (solid red top curve) compared with the theoretical diffraction pattern with the inclusion of the structure factor (solid blue bottom curve) for a Fibonacci grating with $N=300$ lines and $L=23\mu \mathrm{m}$ and $S=15\mu \mathrm{m}$, ${\eta }_3^b=1.875$.