Moiré localization in two-dimensional quasiperiodic systems

We discuss a two-dimensional system under the perturbation of a moiré potential, which takes the same geometry and lattice constant as the underlying lattices but mismatches up to relative rotation. Such a self-dual model belongs to the orthogonal class of a quasiperiodic system whose features have been evasive in previous studies. We find that such systems enjoy the same scaling exponent as the one-dimensional Aubry-André model $\nu \approx 1$, which saturates the Harris bound $\nu >2/d=1$ in two dimensions. Meanwhile, there exists a continuous and rapid change for the inverse participation ratio in the eigenstate-disorder plane, different from the typical one-dimensional situation where only a few or no steplike contours show up. An experimental scheme based on optical lattices is discussed. It allows for using lasers of arbitrary wavelengths and therefore is more applicable than the one-dimensional situations requiring laser wavelengths close to certain incommensurate ratios.

Figure 1

The moiré potential pattern ${\mu }_i$ for (a) the commensurate angle $\theta =arccos(60/61)$ ($q=11,p=1$), and (b) the incommensurate angle $\theta =\pi /5$. Each pixel denotes one lattice site, and the color indicates the strength of the disorder potential therein.

Figure 2

The density of commensurate angles (blue polar lines), for $0\le p\le q\le 20$. Due to mirror symmetry along $x=y$, the angles are symmetric along $\theta =\pi /4$.

Figure 3

The average value for neighboring level gap ratios. Here $\theta =\pi /5$. The numbers of phases ${\phi }_{1,2}$ averaged over are $4000/2000 (L=60), 1000/500 (L=80), 200/100 (L=100), 100/50 (L=120), 60/20 (L=140)$ for $V_d$ within/out-of the range $[3,5]$.

Figure 4

IPR for different eigenstates (arranged by $E_m<E_{m+1}$), averaged over phases ${\phi }_{1,2}$ for each eigenstate. To check possible finite-size effects, we compute two system sizes $L=40$ in (a) and $L=80$ in (b), for the same parameter $\theta =\pi /5$. The green dashed line is the dual point $V_d^{\left(D\right)}=4$.

Figure 5

The scaling of ${\tilde{\alpha }}_0$ at energy $E=0$. The inset is the data collapse of ${\tilde{\alpha }}_0$ as a function of $L/\xi \propto |V_d-V_d^{\left(c\right)}{|}^{\nu }L$ for various system sizes $L$. The number of ${\phi }_1,{\phi }_2$ averaged over are $4\times {10}^4$ for $L=120,140,160,180$ and $2\times {10}^4$ for $L=200,220$, respectively. The stable fit is given by expansion of Eq. (15) to the fifth order. There are totally $N_D=144$ data points, $N_p=8$ fitting parameters, and the goodness of fit is evaluated by ${\chi }^2=134$, with $p$ value $0.52$. (See Appendix pp5-s1 for more fitting details.)

Figure 6

The experimental scheme for the moiré lattice. Black arrows and crosses denote the polarization directions such that the four pairs of laser beams do not interfere with each other. Here $\theta =\pi /3$. The random phases are ${\phi }_1=3,{\phi }_2=2$ chosen as an example.

Figure 7

The portion of particles remaining in odd $x=1,3,5\cdots$. Here $\theta =\pi /3$, and $L=80$.

Figure 8

The (normalized) histogram for ${\tilde{\alpha }}_0$ at $V_d=4.0$ (left), $V_d=4.1$ (middle), and $V_d=4.2$ (right), computed using different ${\phi }_{1,2}$. For larger systems $L$, the distribution inclines towards smaller ($V_d$ in the metal limit) or larger ($V_d$ in the insulator limit) values, while in the critical $V_d$ they overlap.

Figure 9

The histogram for fitted $V_d^{\left(c\right)}$ and $\nu$ at the angle $\theta =\pi /5$, using 1000 synthetic data sets with the same standard deviations as in Fig. 5 at each point. The $95\%$ confidence range is determined by discarding the $2.5\%$ of maximal/minimal data for $V_d^{\left(c\right)},\nu$ on the two sides.

Figure 10

The scaling of ${\tilde{\alpha }}_0$ for $\theta =4\pi /9$ at $E=0$. The inset shows the scaling of ${\tilde{\alpha }}_0$ in terms of $L/\xi \propto |V_d-V_d^{\left(c\right)}{|}^{\nu }L$ after subtracting the scaling-irrelevant term [the second term in Eq. (E2)]. The numbers of ${\phi }_{1,2}$ being averaged over are ${10}^6$ for $L\le 200$ and $4\times {10}^5$ for $L\ge 220$. The fitting data/parameters has $N_D=262,N_p=16$ (with $N_1=N_2=6$), and the quality of fit is evaluated by ${\chi }^2=204,p=0.98$. The uncertainty range is determined by synthetic data set as described previously.