Topological equivalence of crystal and quasicrystal band structures

A number of recent articles have reported the existence of topologically nontrivial states and associated end states in one-dimensional incommensurate lattice models that would usually only be expected in higher dimensions. Using an explicit construction, we here argue that the end states have precisely the same origin as their counterparts in commensurate models and that incommensurability does not in fact provide a meaningful connection to the topological classification of systems in higher dimensions. In particular, we show that it is possible to smoothly interpolate between states with commensurate and incommensurate modulation parameters without closing the band gap and without states crossing the band gap.

Figure 1

The full spectrum of the Aubry-André model (1) with periodic boundary conditions for $b$ from 0 to 2. System size $N=1000$ and the amplitude of the modulating potential $\lambda /t=1$. Note that the spectrum is periodic for $b\to b+1$. With periodic boundary conditions this spectrum has no end states. The black rectangle marks the interval of $b$ that is investigated in detail in the remainder of the article.

Figure 2

Spectra of a finite-length Aubry-André chain as a function of $\varphi$. The chain length is $N=108$. The other model parameters are set at $\lambda /t=1$ and $b=(\sqrt{5}+1)/2$. In the lower panel a potential of the form (2) with $m/t=10$ and $N^{\prime }=106$ has been added to the last two sites near the right end of the chain. End states located on the left are colored blue, and end states on the right are green (see Fig. 3 for an example of such an end state). Only the positions of the right end states (green) are affected by the modified end potential.

Figure 3

Density $|{\psi }_n{|}^2$ as a function of the lattice position $n$ for an end state localized near the right end of the chain. The model parameters have been chosen as in the top panel of Fig. 2. The energy eigenvalue is $E/t=1.04$, placing it in the upper large gap at $\varphi =0$.

Figure 4

Total on-site potential in the Aubry-André model (1) with the additional termination potential (2) at the chain's right end. Parameters have been set at $b=(\sqrt{5}+1)/2$, $\lambda /t=1$, $\varphi =0$, $m/t=4$, $N=12$, and $N^{\prime }=9$.

Figure 5

Spectra of the Aubry-André model with $\lambda /t=1$ and $\varphi =0$ as a function of the inverse period $b$. In the top panel $N=108$; in the bottom panel $N=216$. A potential of the form (2) with $m/t=10$ has been added at the right end of the chain, with $N^{\prime }=99$ (top) and $N^{\prime }=199$ (bottom), effectively limiting the number of lattice sites to 100 and 200, respectively. End states located on the right are colored green.

Figure 6

Spectrum of the Aubry-André model as a function of the inverse period $b$ with parameters $\lambda /t=1$ and $\varphi =0$. At the right end, the chain is terminated with a potential of the form (2) with $m/t=10$. The effective length $N^{\prime }$ is adjusted following the procedure described in the main text, such that $N^{\prime }$ increases from $N^{\prime }=99$ at $b=1.60$ to $N^{\prime }=106.9$ at $b=1.63$ (top panel) as can be seen in detail in Fig. 7, and from $N^{\prime }=199$ at $b=1.60$ to $N^{\prime }=214.9\left(125\right)$ at $b=1.63$ (bottom panel). The total chain length is kept at $N=108$ and $N=216$, respectively. The black lines mark the thresholds of the algorithm described above.

Figure 7

The effective chain length $N^{\prime }$ as a function of the inverse period $b$, for the data set shown in the top panel of Fig. 6.

Figure 8

Spectrum of the Aubry-André model near the second-largest gap in the lower half of the spectrum. Parameters are $\lambda /t=1$, $\varphi =2$, and $m/t=10$. The upper panel shows the spectrum at fixed effective length ($N^{\prime }=199$). In the lower panel the effective length is adjusted following the procedure described in the main text. The effective length $N^{\prime }$ decreases from $N^{\prime }=199$ at $b=1.61$ to $N^{\prime }=183.08$ at $b=1.62$. End states located near the right end are shown green. The black lines mark the thresholds of the algorithm described above.

Figure 9

Spectra of the Aubry-André model as a function of $\varphi$, for $\lambda /t=1$ and for different values of the inverse period $b$, $b=1.6014875$, $1.6017375$, $1.6019875$, $1.6021125$, $1.6023625$, $1.6026125$, $1.6029875$, and $1.6032375$ from top left to bottom right. A potential of the form (2) with $m/t=10$ is included and the effective chain length $N^{\prime }$ is determined by application of the procedure of Sec. 3 to the case $\varphi =0$ and ranges from $N^{\prime }=99.1$ (at $b=1.6014875$) to $N^{\prime }=99.8375$ (at $b=1.6032375$). End states located on the left end are colored blue, states at the right are shown green.