Classification of Fragile Topology Enabled by Matrix Homotopy

TL;DR

The paper addresses fragile topology in moiré materials, which challenges momentum-space classifications in finite, disordered, or aperiodic systems. It introduces an energy-resolved $\mathbb{Z}_2$ marker based on matrix homotopy by mapping the protecting $C_2\mathcal{T}$ symmetry to matrix relations and using the spectral localizer $L_{(x,y,E)}$, with the invariant $\zeta_E = \mathrm{sign}(\mathrm{Pf}(L_{(0,0,E)}))$ and a robustness measure via the local gap $\mu_{(0,0,E)}$. The approach is demonstrated on a disordered $C_2\mathcal{T}$-symmetric TBG model and a 2D photonic crystal, revealing disorder-induced re-entrant fragile topology and applicability to gapless heterostructures without requiring eigenstate solutions. This framework provides a general, finite-system, position-space tool for identifying fragile topology across platforms, with potential impact on topological photonics and correlated electron phenomena.

Abstract

The moire flat bands in twisted bilayer graphene have attracted considerable attention not only because of the emergence of correlated phases but also due to their nontrivial topology. Specifically, they exhibit a new class of topology that can be nullified by the addition of trivial bands, termed fragile topology, which suggests the need for an expansion of existing classification schemes. Here, we develop a Z2 energy-resolved topological marker for classifying fragile phases using a system's position-space description, enabling the direct classification of finite, disordered, and aperiodic materials. By translating the physical symmetries protecting the system's fragile topological phase into matrix symmetries of the system's Hamiltonian and position operators, we use matrix homotopy to construct our topological marker while simultaneously yielding a quantitative measure of topological robustness. We show our framework's effectiveness using a C2T-symmetric twisted bilayer graphene model and photonic crystal as a continuum example. We have found that fragile topology can persist both under strong disorder and in heterostructures lacking a bulk spectral gap, and even an example of disorder-induced re-entrant topology. Overall, the proposed scheme serves as an effective tool for elucidating aspects of fragile topology, offering guidance for potential applications across a variety of experimental platforms from topological photonics to correlated phases in materials.

Figures (3)

• Figure 1: (a) Bulk band structure for the $C_2\mathcal{T}$-symmetric TBG model. Inset diagrams this lattice model. (b) The local gap $\mu_{(x,y,E)}$ for $x$ and $E$ at $y = 0$. Position $x$ is scaled in terms of the lattice constant $a$ and $x_m$ denotes the length of the edge from the origin. (c) $\mu_{(x,y,E)}$ for $E$ at the center of rotation denoted by blue dotted line in panel (b). The energy-resolved invariant $\zeta_E$ is shown by the green dots, where $\kappa = 0.1 t/a$ for all calculations.
• Figure 2: (a) Closings of $E_{\textrm{gap}}$ due to uniform changes in $\phi$ in the periodic $C_2\mathcal{T}$-symmetric TBG model. (b) Spectrum of $L_{(0,0,0)}$ (gray) and $\zeta_0$ (green) at the middle of the bulk band gap as $\phi$ is uniformly varied in the finite TBG system. (c) Hopping phases of clean and disordered systems depicted on the unit circle. The red and blue dots correspond to the opposite off-diagonal terms of the Hamiltonian, forming conjugate pairs. (d), (e), (f) Ensemble analysis of $E_{\textrm{gap}}$ (d), $\mu_{(0,0,0)}$ (e), and histogram of $\zeta_{0}$ for the 10 disorder configurations for increasing $S$ (f). In (d) and (e), the black dashed lines show the behavior for uniform hopping phase changes in the clean system. Solid Gray lines show the results for each disorder configuration, the solid green line shows the average over the ensemble, and the gray shading fills the area between the maximum and minimum of these data, representing the sample deviations. The histogram in (f) uses green and gray to indicate $\zeta_0=-1$ and $+1$, respectively.
• Figure 3: (a) Schematics of a single unit cell of the photonic crystal (upper) and a heterostructure (lower) formed by $6\times6$ unit cells surrounded by air bounded by a perfect electrical conductor (PEC). Here, $\epsilon_1$ ={16, 6$i$, 0; $-$6$i$, 16, 0; 0, 0, 16}, $\epsilon_2=4$, $\epsilon_3=1$, $a$ is the lattice constant, $r = 0.2a$, and $d = 0.45a$. (b) Band structure of the bulk photonic crystal. (c),(d) The density of states (DoS) (c) and the local gap and frequency-resolved index (d) for the $6\times6$ finite system from (a). Calculations in (d) use $\kappa$ = 0.01 $2\pi c/a^2$, where $c$ is the speed of light in a vacuum. (e) Local density of states (LDoS) within the fragile band gap depicted by green in (b) at the two frequencies for the finite heterostructure shown in (a).

Theorems & Definitions (10)

• Remark SI.1
• Definition SI.2
• Lemma SI.3
• proof
• Theorem SI.4
• proof
• Theorem SI.5
• proof
• Remark SI.6
• Example SI.7