Physical Properties of an Aperiodic Monotile with Graphene-like Features, Chirality, and Zero Modes

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including sixfold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette ($\varphi$) is half of the flux quantum, zero modes are found localized around the reflected “anti-hats”; and (v) its Hofstadter spectrum is periodic in $\varphi$, unlike for other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

Figure 1

Spectral properties of the vertex model Eq. (1) on the Hat. (a) $H2$, with 22 anti-hats (reflected images of the hat tile) shown in blue. The inset shows a patch of the Hat quasilattice (solid circles) overlayed with a graphene hexagonal lattice approximant (open circles). (b) Density of states of the vertex model Eq. (1) on $H2$. (c) Momentum-resolved spectral function ${\mathcal{A}}_{+}(E,\mathbf{k})$ of the enantiomer in (a) along the $k_x$ momentum direction, calculated using the Kernel polynomial method [52]. The dispersion relation Eq. (2) for the hexagonal lattice [inset of (a)], Eq. (2), is overlaid in orange, with parameters $(a_g,t_1,t_2,{ϵ}_0)=(2a/\sqrt{3},0.82,-0.025,-0.2)$. (d) ${\mathcal{A}}_{+}(E=-0.2,\mathbf{k})$ as a function of momentum $\mathbf{k}=(k_x,k_y)$. (e) Difference between the spectral functions ${\mathcal{A}}_{\pm }(E=-0.2,\mathbf{k})$ of the system in (a) and its reflection in the $y$ axis.

Figure 2

Zero modes under 0 and $\pi$ flux. (a) Low-energy spectrum of the Hat quasilattice without flux. The eight exact zero modes are shown in red. (b) The associated local density of states (LDOS) of these zero modes. Colors highlight previous inflation generations ($T1$, $P1$, $F1$, and $H1$) composing $H2$. (c) The $T1$ quasicrystallite has no zero modes. (d) The $T1$ quasicrystallite without the rightmost hat has a single zero mode. The overlaid zero-mode amplitudes form the Sutherland loop sequence $\{0,1,0,-1{\}}^m$ [64] of length $4m$, with $m\in \mathbb{Z}$. (e) The low-energy spectrum under $\pi$ flux. Red: the 22 exact zero modes. (f) Corresponding zero-energy LDOS, pinned to anti-hats. (g) Zoom of the wave function corresponding to eigenstate 530. Because $\varphi /{\varphi }_0=1/2$, the Sutherland loop is modified to have one defect per anti-hat.

Figure 3

Hofstadter spectrum and quantized conductance. (a) The bulk Hofstadter spectrum Eq. (1) calculated for $H3$ near the band bottom. (b) The local Chern marker ${\mathcal{C}}_{\mathbf{r}}$ calculated for $H3$ (black) at $E=-1.58$ and $\varphi /{\varphi }_0=0.2$. The yellow dashed line delimits the bulk area ${\mathcal{A}}_b$ used to produce panels (a) and (c). Inside ${\mathcal{A}}_b$, the marker averages to $\mathcal{C}=-1.03$. Thick black lines on the edges denote sites where leads are attached. (c) $\mathcal{C}$ as a function of $E$ and $\varphi /{\varphi }_0$. Bulk gap regions of panel (a) have nontrivial Chern numbers. (d) Corresponding two-terminal conductance map. Where $|\mathcal{C}|=1$, the conductance is quantized to $G=G_0=e^2/h$, confirming the role of topological boundary modes in transport. The leads can be placed arbitrarily so long as they are well separated.