Topological Gyromorphs

Abstract

Gyromorphs are a new class of disordered systems that combine an amorphous-like absence of translational order with quasi-long-range rotational order. Gyromorphs can outperform quasicrystals or hyperuniform arrangements in forming isotropic band gaps, suggesting an avenue to realize robust disordered topological phases. However, gyromorphs lack exact rotational symmetry, which is only realized on average, posing an obstacle for existing real-space invariants to correctly diagnose topological gyromorphs. In this work we show that gyromorphs can host higher-order topological insulating (HOTI) phases protected by average rotational symmetry, and we develop and systematically compare tools for diagnosing topological phases protected by such symmetry. We introduce symmetry indicators of the effective Hamiltonian based on average rotational symmetries which, when combined with the spectral localizer and a scattering invariant, draw a consistent topological phase diagram. Our work unlocks gyromorphs as a novel platform to study topological phases beyond crystals, quasicrystals, and amorphous materials.

Figures (5)

• Figure 1: (a) Real-space representation of the $C_8$ locally symmetric gyromorph lattice with $N=5 \times 10^3$ sites. (b) Reciprocal-space structure factor of the gyromorph lattice with $N=10^4$. The momentum axes are expressed in units of the average lattice spacing $a= 1/\sqrt{N}$. (c) Bottom: 100 smallest eigenvalues of the real-space Hamiltonian \ref{['eq:hamgen']}, defined on the same lattice as in (b), as a function of the on-site parameter $\mu$. Violet and gray points denote corner and bulk states, respectively. Shading indicates the putative topological region. Top: The local density of states (LDOS) of the zero-energy modes is shown for $\mu/t=-2$.
• Figure 2: (a) Norm of the commutator $[H_{\mathrm{eff}}(\mathbf{k}), C_8 M]$ from \ref{['eq:effectiveH', 'eq:cnm']} in momentum space. (b) Minimum bulk gap of the effective Hamiltonian in momentum space for $\mu/t \in [-20, 5]$, in logarithmic scale. The points $\Gamma$ and $A$ correspond to the high-symmetry momenta where the bulk gap closes. (c) Spectrum of the effective Hamiltonian computed at the $\Gamma$ and $A$ points represented by solid and dashed lines, respectively, as a function of the mass parameter $\mu / t$. The color encodes the $C_8M$ eigenvalues associated with each subspace. We see two crossings as a function of $\mu/t$, each involving a different pair of symmetry eigenvalues.
• Figure 3: Topological phase diagram as a function of the staggered mass $\mu/t$. (a) Real-space Hamiltonian spectrum from \ref{['fig:Fig1']}(c). (b) Inversion of the $C_8 M$ eigenvalues at the gap closings of the effective Hamiltonian at the high-symmetry momenta from \ref{['fig:Fig2']} (c) Half-signature of symmetry reduced localizer from \ref{['eq:idxlocalizer']} for $\kappa=0.5$, and $\mathbf{x}=(x_0, y_0)$ shown in the inset. Green and red circles denote the positive and negative chiral charges of the corner modes. The numerical instability at $\mu/t\sim 0$ is consistent with a small gap of both the Hamiltonian and spectral localizer, see Appendix \ref{['App:reducedlocalizer']}. (d) Higher-order scattering invariant from \ref{['eq:scattering_invariant']}, computed using the eight-fold symmetric transport setup shown in the inset.
• Figure S1: (a) Chiral charges of the zero-energy modes for $\mu/t \sim -2$, green and red circles denote positive and negative chiral charges, respectively. The vector $\hat{e}_d$ connects two opposite vertices of the octagon, while $\hat{e}_x$ is the horizontal direction. We mark the point $x_0, y_0$ where we compute the invariant in \ref{['fig:Fig3']}(c) in the main text. (b) Half-signature of the symmetry-reduced localizer as a function of $x$, along direction $\hat{e}_d$ in pink and $\hat{e}_x$ in purple.
• Figure S2: (a) Half-signature of the symmetry-reduced localizer as a function of $\mu/t$ from \ref{['fig:Fig3']}(c), with $\kappa=0.5$, $\mathbf{x}=(x_0, y_0)$. (b) Local gap of the symmetry-reduced localizer corresponding to panel (a) as a function of $\mu/t$. The shaded region is that shown in \ref{['fig:Fig3']}.