Probing Topological Stability with Nonlocal Quantum Geometric Markers

Abstract

Spatially resolved local quantum geometric markers play a crucial role in the diagnosis of topological phases without long-range translational symmetry, including amorphous systems. Here, we focus on the nonlocality of such markers. We demonstrate that they behave as correlation functions independently of the material's structure, showing sharp variations in the vicinity of topological transitions and exhibiting a unique pattern in real space for each transition. Notably, we find that, even within the same Altland-Zirnbauer class, distinct topological transitions generate qualitatively different spatial signatures, enabling a refined, class-internal probe of topological stability. As such, nonlocal quantum geometric indicators provide a more efficient and versatile tool to understand and predict the stability of topological phase transitions.

Figures (3)

• Figure 1: Nonlocal quantum geometry at the topological phase transition for the crystalline system. (a) Energy spectrum as a function of $M/t$. Gap closings occur at $M/t = 0, -4, -8$, indicating topological phase transitions. (b) Chern number $C$ and quantum metric $\Omega$. The finite width of the phase transitions stem from the finite size of the sample. (c-d) Real-space patterns of the nonlocal Chern marker $C(r,r_0)$ (c) and quantum metric $\Omega(r,r_0)$ (d) close to each topological transition: ($i$) $M/t=-7.8$, ($ii$) $\ M/t = -4$, and ($iii$) $\ M/t = -0.2$. Solid, dashed, and dashed-dotted vertical lines and frame contours indicating the different $M$ values. Crystalline lattice is drawn for reference and black cross indicates reference site $r_0$.
• Figure 2: (a) Topological phase diagram of the crystalline model as a function of $M/t$ and disorder strength $\delta M/t$. (b) Topological markers $|C_\mathbf{k}|$ for $\mathbf{k} = (0,0)$ (red), $\mathbf{k} = (\pi,0)$ (green) and $\mathbf{k} = (\pi,\pi)$ (in blue), as a function of $M/t$ and disorder strength $\delta M/t$. (c) Markers of nonlocal quantum metric $\Omega_\mathbf{k}$ for $\mathbf{k} = (0,0)$ (pink), $\mathbf{k} = (\pi, 0)$ (gold) and $(\pi,\pi)$ (green). In (b,c), color intensity represents the absolute value of the indicator, each with their own hue.
• Figure 3: Nonlocal quantum geometry at the topological phase transitions for the amorphous system. (a) Energy spectrum as a function of $M/t$. Compared to the crystalline case (Fig. \ref{['fig:crystal']}), the two topological transitions at $M = -8$ and $M=-4$ are merged. (b) Local Chern marker $C$ and quantum metric $\Omega$ averaged over the central region. (c-d) Real-space patterns of the nonlocal Chern marker $C(r,r_0)$ (c) and quantum metric $\Omega(r,r_0)$ (d) close to each gap closing: ($i$) $M/t = -6$, ($ii$) $M/t = -5$, and ($iii$) $M/t = -0.2$. Solid, dashed, and dashed-dotted vertical lines and frame contours indicate the different $M$ values. Amorphous lattice is drawn for reference and black cross indicates reference site $r_0$.