Rational approximations of quasiperiodicity via projected Green's functions

We introduce the projected Green's function technique to study quasiperiodic systems such as the Aubry-André-Harper (AAH) model and beyond. In particular, we use projected Green's functions to construct a rational approximate sequence of transfer matrix equations consistent with quasiperiodic topology, where convergence of these sequences corresponds to the existence of extended eigenfunctions. We motivate this framework by applying it to a few well-studied cases such as the almost-Mathieu operator (AAH model), as well as more generic no-dual models that challenge standard routines. The technique is flexible and can be used to extract both analytic and numerical results, e.g., we analytically extract a modified phase diagram for Liouville irrationals. As a numerical tool, it does not require the fixing of boundary conditions and circumvents a primary failing of numerical techniques in quasiperiodic systems—extrapolation from finite size. Instead, it uses finite-size scaling to define convergence bounds on the full irrational limit.

Figure 1

Magnetic unit cells are chosen in the 2D parent Hamiltonian (center), but not all magnetic unit cells have a convergent rational approximate sequence, see Appendix pp4. The convergent alignment defines allowed projections back to 1D (left/right), i.e., horizontal unit cells naturally project back to 1D, while vertical unit cells do not. Concretely, for the AAH model, Eq. (3) the regimes $V<t$ and $V>t$ dictate a horizontal or vertical cell, respectively. This is rooted in the topology of the 2D parent system that defines the projection to the 1D quasiperiodic model.

Figure 2

Numerically computed AAH spectrum (green), corresponding spectral gaps (grey), and pGf zeros (red x) for parameters $t=1,V=0.5,N=2048,\alpha =\frac{1}{2}(\sqrt{5}-1)$. Each point is labeled by its transfer matrix eigenvalues. Spectrum (green) has transfer matrix eigenvalues on the unit circle, $|{\lambda }_T|=1$. Zero eigenvalues of the pGF (red x) imply the transfer matrix is rank deficient, ${\lambda }_T=0$. Note the many small gaps (grey) and zeros in almost (up to numerical precision) every gap.

Figure 3

Plot of max (operator norm) difference over all $\omega$ between the rational and irrational pGfs for $\alpha =(\sqrt{5}-1)/2$. Convergence is defined by the ability to take $\left|\right|G_{\alpha }(\omega +i\epsilon )-G_N(\omega +i\epsilon )\left|\right|<{\epsilon }^2.$ For $V/t<1$ ($V/t>1$), the horizontal (vertical) unit cell converges. Taking larger system sizes $N$ sharpens convergence/divergence (shrinking divergence for large $V/t$ caused by finite system size_ . Inset: AAH phase diagram for Liouville $\alpha$, with $\beta$ the Liouville exponent and transition at $\beta =ln\left(V\right)$ (blue: metal, red: insulator, gray: transition).

Figure 4

Convergence criteria versus the inverse participation ratio (IPR) as measure of eigenfunction localization versus delocalization. Panel (a) shows max criteria of convergence versus inset max IPR. Panel (b) displays average criteria versus inset mean IPR. For non-self-dual models, mobility edges persist deep into the metallic regime (Fig. 5). Note, for small $N$, mean IPR doesn't converge to 1 as quasiperiodicity breaks down.

Figure 5

The pGf convergence and the IPR for the non-self-dual potential in Eq. (18) plotted on the same axis ($L=2048, V=1/2$). We notice the exact alignment between the two measures of localization. Both localized and extended solutions appear depending on $\omega$, with a clear transition at ${\omega }_{*}\sim \pm 1.4t$, a mobility edge. We emphasize that relative values are not comparable – the pGf cutoff is ${10}^{-2}$ and the IPR is a measure normalized between $1/L$ and 1. Isolated poles inside gaps correspond to nonuniversal edge modes, artifacts of finite-size simulation.

Figure 6

(a) Quasiperiodic spectrum as function of plaquette flux $\mathrm{\Theta }=2\pi \alpha$. Also note the gap labeling index corresponds to slope of line $y=m\alpha +n$ with $m,n\in \mathbb{Z}$. (b) Illustration of quasiperiodic pattern generating a minimal surface (hull). This forms the underlying unital algebra, taking the place of a Brillouin zone.

Figure 7

Plot of pGf convergence for GAAH model at $V=0.7,t=1$ for $b=1/2,1/4$ and $N=4096$. We notice the shift in mobility edge from convergence criteria. Note, blue/yellow lines indicate analytic mobility edge, and blue/yellow boxes indicate spectral gaps in which those mobility edges line. The blue line occurs deep in a spectral gap for which states above are localized and states below are delocalized, while the yellow line is closer to the gap edge. Since the the quasi-periodic spectrum forms a cantor set, all mobility edges will fall in a spectral gap, but numerical precision of pGf convergence is better in large gaps.