Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal

One-dimensional (1D) quasicrystals exhibit physical phenomena associated with the 2D integer quantum Hall effect. Here, we transcend dimensions and show that a previously inaccessible phase of matter—the 4D integer quantum Hall effect—can be incorporated in a 2D quasicrystal. Correspondingly, our 2D model has a quantized charge-pump accommodated by an elaborate edge phenomena with protected level crossings. We propose experiments to observe these 4D phenomena, and generalize our results to a plethora of topologically equivalent quasicrystals. Thus, 2D quasicrystals may pave the way to the experimental study of 4D physics.

Figure 1

(a) An illustration of $H$ [cf. Eq. (1)]: a 2D square lattice with a cosine-modulated on-site potential. The potential is incommensurate with the lattice, and is $\sigma$ dependent. The state $\sigma =+$ experiences the upper (blue) potential, and $\sigma =-$ experiences the lower (red). For clarity, the potentials are vertically displaced. The dots mark the underlying lattice sites. (b)–(d) A quantized charge pumping along the $x$ direction is achieved by scanning the shift parameters ${\varphi }_x$ and ${\varphi }_y$ in the presence of the modulation modifications ${\overline{B}}_{yz}$ and ${\overline{B}}_{wy}$, respectively [cf. Eqs. (5, 7)]. The effects on the potentials of $\sigma =+$ (blue) and $\sigma =-$ (red) is illustrated [the thin (gray) line denotes the unmodified reference potential]: (b) ${\varphi }_x$ (or, equivalently, ${\varphi }_y$) shifts the cosine potentials in opposite directions for opposite $\sigma$. (c) ${\overline{B}}_{yz}$ makes the modulation frequency become $\sigma$ dependent $b_y+\sigma {\overline{B}}_{yz}$. (d) ${\overline{B}}_{wy}$ shifts the cosine potentials in opposite directions, but with an increasing shift along the $y$ direction.

Figure 2

The spectrum of $H$ as a function of the shift parameter ${\varphi }_x$ [cf. Eq. (1)], for an open $x$ coordinate and a periodic $y$ coordinate. The horizontal bands correspond to bulk states. The gap-traversing states with $\sigma =+$ (blue) or $\sigma =-$ (red) are edge states, which are localized at the left or right edges (see insets for typical wave functions). The crossings of edge states are topologically protected. The values of the 2nd Chern number, $\mathcal{V}$, associated with the large gaps are presented.