Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor

Abstract

The strict geometric rules that define aperiodic tilings lead to the unique spectral and transport properties of quasicrystals, but also limit our ability to design them. In this Letter, we explore a novel example of a continuously tunable family of two-dimensional aperiodic tilings in which the underlying real-space geometry becomes a control knob of the wavefunction's quantum geometric tensor. The real-space geometry can be used to tune into topological phases occupying an expanded phase space compared to crystals, or into a disorder-driven topological Anderson insulator. The quantum metric can also be tuned continuously, opening new routes towards tunable single- and many-body physics in aperiodic solid-state and synthetic systems.

Figures (10)

• Figure 1: Tiles and tilings for different values of the tile side length, defined by two side lengths, $\ell$ and $1-\ell$. (a) shows a selection of tiles, including the Chevron, Hat, Turtle, and Comet. Their corresponding aperiodic tilings are shown in (b-e). For each $\ell$ we place two orbitals $l$ at every vertex, with onsite terms $\epsilon_{ll'}$. Pairs of sites $\{\alpha,\beta\}$ are connected by the space and orbital dependent hopping $T_{ll'}(\mathbf{r}_{\alpha\beta})$, see Eq. \ref{['eq:hamiltonian']}.
• Figure 2: Spectral, topological, and geometric properties of the model for different values of $\ell$. (a) Energy spectrum of the bulk Hamiltonian as a function of $\ell$. We obtain the bulk spectrum by excluding states whose weight inside the yellow contour in (d) and (e) is smaller than their weight outside it. Magenta dashed lines indicate the $\ell$ values used in panels (b) and (c). The yellow dashed line indicates the Fermi energy $E_F=-1.15t$. (b) Spectral function $\mathcal{A}(E,{\bf p})$ for $\ell=0.23$, showing a full gap near 1/4 filling, attained near the energy indicated by the white dashed line. (c) Spectral function $\mathcal{A}(E,{\bf p})$ for $\ell=1/3$, showing the presence of edge states within bulk gaps. (d) Local Chern marker $\mathcal{C}_{{\bf r}_i}$ Eq. \ref{['eq:LCM']}, evaluated at the energy indicated by the white dashed line in (c) for $\ell=1/3$. The yellow solid line outlines the bulk region used to perform the averaging. In this region, the average value of the local Chern marker is $0.98$. (e) Local quantum metric $g_{\bf r}$, Eq. \ref{['eq:QM_marker']}, computed at the $E_F$ indicated by the dashed line in (c) for $\ell=1/3$. We observe that the local quantum metric takes large values at the edges, indicating delocalized edge states. In all the panels the system has $2530$ sites and $M/t=-2.7$. In panels (b) and (c) we used the kernel polynomial method Wei_KPM_2006 to compute the spectral function with 512 moments.
• Figure 3: Topological phase diagram. (a) Bulk half-signature of the spectral localizer at $E_0=E_F$ as a function of both the geometrical parameter $\ell$ and the value of the onsite term $M/t$. $E_F$ is fixed at $3/4$ filling out of a total of $N_\mathrm{sites}\times N_{\mathrm{orbs}}=2530\times 2$ states. We observe topological phases for a wider parameter region compared to the square-lattice crystalline model, whose phase boundaries are delimited by the black dashed lines. (b) Localizer gap corresponding to (a). The gap in the spectrum of the spectral localizer closes at the boundary between the topological and trivial phases of panel (a). (c) Bulk half-signature of the spectral localizer averaged over 100 disorder realizations at $E_0=E_F$ and $M/t = 0.5$, as a function of $\ell$ and onsite disorder strength $W/t$. We observe two types of behavior as we increase $W/t$ for fixed $\ell$: either the clean system is topological and transitions into a trivial Anderson insulator (e.g. at $\ell=0.5$), or a trivial clean system transitions into a topological phase, to later become a trivial Anderson insulator (e.g. at $\ell=0.8$ or $\ell=0.2$). (d) The localizer gap vanishes between the topological and trivial Anderson insulators in (c). In all panels, the system has $N_\mathrm{sites}=2530$ sites and $\kappa = 0.01$. Panels (b) and (d) are plotted using a logarithmic colormap.
• Figure 4: (a) Energy spectrum as a function of $\ell$, showing a zoomed-in version of Fig. \ref{['fig:spectral_function']}a including bulk and edge states. $3/4$ filling is shown as a yellow dashed line in (a). (b) Bulk trace of the local quantum metric $G$ and local Chern marker $C$ for this filling as a function of $\ell$. At small $\ell$ the sites cluster and localize the wavefunctions for this filling, leading to a small $G$ with $C\approx 0$. At intermediate $\ell \approx 0.5$, $G\ge C \approx 1$, satisfying the trace inequality. For large $\ell \gtrsim 0.7$ this filling sits in the middle of a band of delocalized states with increased $G$. In both panels, the system has $N_\mathrm{sites}=2530$ and $M/t=-2.7$.
• Figure S1: Tile and inflation rule. (a) A tile is a fourteen-sided polygon with edge lengths parameterized by a geometric parameter $\ell$. In this example, the value of $\ell$ is taken to be $1/(1+\sqrt{3})$. (b) To construct the tiling, we use clusters of tiles. We begin by defining a two-tile compound consisting of one tile (in white) and its mirror image (in pale blue). We then establish a substitution rule for both the single tile and the two-tile compound using the H7 and H8 clusters.