Topological Weaire-Thorpe models of amorphous matter

Abstract

Amorphous solids remain outside of the classification and systematic discovery of new topological materials, partially due to the lack of realistic models that are analytically tractable. Here we introduce the topological Weaire-Thorpe class of models, which are defined on amorphous lattices with fixed coordination number, a realistic feature of covalently bonded amorphous solids. Their short-range properties allow us to analytically predict spectral gaps. Their symmetry under permutation of orbitals allows us to analytically compute topological phase diagrams, which determine quantized observables like circular dichroism, by introducing symmetry indicators in amorphous systems. These models and our procedures to define invariants are generalizable to higher coordination number and dimensions, opening a route toward a complete classification of amorphous topological states in real space using quasilocal properties.

Keywords: amorphous solids; symmetry indicators; topological phases.

Fig. 1.

Topological Weaire–Thorpe models. (A) Basic building blocks for threefold, fourfold, fivefold, and sixfold coordinated Weaire–Thorpe models. The diagrams show a single site $i$ with $z$ orbitals labeled by $j$. The green lines indicate the intrasite hopping $V{e}^{i\varphi }$, while the blue lines are the intersite hopping $W$. (B) Example of a threefold coordinated topological Weaire–Thorpe model.

Fig. 2.

Spectral properties. (A) The energy spectrum for $W/V=0.66$ as a function of $\varphi$ with periodic boundary conditions. The color intensity is proportional to the density of states (DOS). The dashed and dotted lines are obtained from the resolvent inequalities Eq. 4, and correspond to states with $F_{+}=1$ and $F_{-}=1$, respectively (Fig. 3D). Their color coding follows that of Fig. 3C. The vertical line $\varphi =1.3$ indicates the parameters chosen for B–D. (B) The color bar shows the participation ratio $p={\left({\sum }_i{\left|{\psi }_i\right|}^2\right)}^2/N{\left|{\psi }_i\right|}^4$ for periodic boundary conditions, a measure of localization that indicates the ratio of sites contributing to the density of states within a given energy bin. The height of the histogram is proportional to the density of states. (C) The local density of states with open boundary conditions for an in-gap state at 2/3 filling indicated by the black line in B, showing the edge support of in-gap states. (D) The local Chern marker density $c\left(\mathbf{r}\right)$ at 2/3 filling, quantized to $C=-1$ in the bulk, with a large and positive edge contribution, typical of a Chern insulating phase. The white dashed square shows the averaging region used to compute Fig. 3A.

Fig. 3.

Topological phase diagram and symmetry properties of the $z=3$ Weaire–Thorpe model at $\nu =2/3$ filling. (A) Topological phase diagram obtained from the local Chern marker density averaged over the area within the dashed white square in Fig. 2D, ${⟨c\left(\mathbf{r}\right)⟩}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$. The solid lines indicate the gap closing transitions obtained using the inequalities Eq. 4. The vertical dashed line indicates $\varphi =1.3$, used in C and D. (B) Topological phase diagram using the symmetry indicator formula Eq. 6. (C) Spectral densities $F_m\left(E\right)$. We use an RGB color value to visualize how a given eigenstate transforms under $C_3$ rotations (see Symmetry Indicators and Topological Invariants). The dotted lines show the effective Hamiltonian spectrum at $\mathbf{k}=0$ for $l=0,\pm 1$ with the same color coding. The gray dotted line indicates a 2/3 filling. The vertical dashed lines indicate $W=\mathrm{0,2}$, $W=\mathrm{0,3}$, $W=\mathrm{0,4}$ used in E. The Lower Left schematic shows the trivial decoupled triangle limit ($W=0$). (D) Spectral densities $F_{\pm }\left(E\right)$. We use a two-color coding to visualize how a given eigenstate transforms under bond inversion (see Symmetry Indicators and Topological Invariants). The Lower Right schematic shows the trivial dimer limit ($V=0$). (E) Momentum resolved spectral weights $F_m\left(E,\mathbf{k}\right)$ showing a band inversion at $\left|\mathbf{k}\right|\equiv k=0$. The eigenvalues of $H_{\mathrm{e}\mathrm{ff}}\left(\mathbf{k}\right)$ are shown as dark dotted lines. The red, green, and blue colors correspond to $m=\mathrm{0,1},-1$ respectively. The continuum Chern number $\nu$ changes from $\nu =0$ to $\nu =-1$ across the transition.