Universal Boundary-Modes Localization from Quantum Metric Length

TL;DR

This work uncovers a universal link between boundary-mode localization in topological flat-band systems and the quantum metric, introducing the quantum metric length (QML) as a lower bound for boundary-mode spread in the flat-band limit. By engineering degenerate flat bands with tunable quantum geometry through a subdominant coupling, the authors show TBMs naturally exhibit two decay phases: a conventional dispersion-driven decay and a geometry-driven exponential decay, with the QML controlling long-range localization via the non-Abelian quantum metric. They validate the framework in concrete lattice models (e.g., variant Lieb and Lieb-QWZ), highlighting a hotspot in the Brillouin Zone near the $M$-point that amplifies the QML and yields anisotropic, tunable boundary localization. The work connects these geometric effects to observable transport phenomena—quantum Hall plateau shaping and Fraunhofer-pattern crossover in flat-band Josephson junctions—suggesting moiré and other engineered platforms as viable routes to harness QML for boundary-mode engineering and novel quantum transport.

Abstract

The presence of localized boundary modes is an unambiguous hallmark of topological quantum matter. While these modes are typically protected by topological invariants such as the Chern number, here we demonstrate that the {\it quantum metric length} (QML), a quantity inherent in multi-band topological systems, governs the spatial extent of flat-band topological boundary modes. We introduce a framework for constructing topological flat bands from degenerate manifolds with large quantum metric and find that the boundary modes exhibit dual phases of spatial behaviors: a conventional oscillatory decay arising from bare band dispersion, followed by another exponential decay controlled by quantum geometry. Crucially, the QML, derived from the quantum metric of the degenerate manifolds, sets a lower bound on the spatial spread of boundary states in the flat-band limit. Applying our framework to concrete models, we validate the universal role of the QML in shaping the long-range behavior of topological boundary modes. Furthermore, by tuning the QML, we unveil extraordinary non-local transport phenomena, including QML-shaped quantum Hall plateaus and anomalous Fraunhofer patterns. Our theoretical framework paves the way for engineering boundary-modes localization in topological flat-band systems.

Figures (10)

• Figure 1: An illustration of two paradigms in constructing 2D topological flat bands. Upper panel: Decoupled atomic orbitals form degenerate flat bands with trivial quantum metric, resulting in well-localized TBMs with zero QML after topological transition. Lower panel: Degenerate flat bands with nontrivial quantum metric, after the same process, leads to geometrically nontrivial TBMs with long-range localization controlled by the large QML.
• Figure 2: Band structure, Topology and QML of the Lieb-QWZ model. (a) Band structure of the Lieb-QWZ model. (b) Zoomed-in view of the topological flat bands. (c) The Chern number evolution of a flat band as a function of the mass term $m/t$. (d) The BZ distribution of the traced non-Abelian quantum metric tensor $\bar{\mathcal{G}}_{\bm{k}} \equiv \frac{1}{N_f}\sum^{N_f}_{\alpha\alpha'} \operatorname{Tr}[{\mathcal{G}_{ab}^{\alpha\alpha'}}(\bm{k})]$ of the exactly degenerate flat bands, featuring a hot spot at the $M$-point. The green (purple) dashed line represents $k_x=0$ ($k_y=0$) cut lines respectively. (e) The $\bar{\mathcal{G}}_{M}$ as a function of $\delta$. (f) The $x$($y$)-directional QML $\xi_{QM,x,k_y}$ ($\xi_{QM,y,k_x}$) around the $\Gamma$-channel. Parameters: $(m,\tilde{t},t,\alpha,\delta)=(4\times 10^{-4},2\times 10^{-4},2\times 10^{-4},0.1,0.05)$ for (a) to (c); $(m,\tilde{t},t,\alpha,\delta)=(0,0,0,0.5,0.01)$ for (d) to (f). $J=1$ for all panels.
• Figure 3: Spatial behaviors of TBMs in the Lieb-QWZ model. (a,b,e,f) The spectrum for the Lieb-QWZ model with OBC (a) in the $x$-direction and (e) in the $y$-direction. (b) and (f) shows the corresponding flat bands zoomed in, with in-gap modes highlighted in red and blue for the left and right branches, respectively. (c,g) The logarithmic wave functions for different $\delta$ with OBC in (d) the $x$-direction and (h) the $y$-direction. To manifest both behaviors, we keep the parameters the same as before except $t = 0.5\tilde{t}$. (d) The logarithmic wave functions for different $\tilde{t}/t$ with OBC in the $x$-direction (vertically shifted for visual clarity). (h) In the case $t=\tilde{t}$, the ratio $\Omega_{x,k_y}/a\xi_{QM}$ (where $a$ is the lattice constant) is consistently bounded below by $1$ in the Lieb-QWZ model, in excellent agreement with Eq. (\ref{['Eq_inequality_LocalizationFunc_ge_nonAbelian_QM']}). Parameters: $t = \tilde{t}$ for (a,b,e,f), and $J=1$, $\tilde{t}= 2\times10^{-4}J$, $m = 2\tilde{t}$, $\delta = 0.05$ and $\alpha = 0.1$ for all panels.
• Figure 4: Quantum Hall resistivity and the QML. (a) A 6-terminal Hall bar device. The green dashed lines represent inter-edge scattering of the chiral edge states due to a large QML. (b) The Hall resistivity $R_H$ (solid lines) and the QML $\xi_{QM,x,k_y}$ (dash-dot lines) versus the chemical potential $\mu/t$ for different $\delta$. The chemical potential in the lower axis is locked with the momentum in the upper axis due to the linear dispersion of TBMs. The width for the device and the leads is fixed at $W_1=30a$, and the spatial separation of the terminals is set as $W_2=400a$. Other parameters are the same as Fig. \ref{['Fig_Lieb_QWZ_OBC_spectrum_wavefunc_QSQM']}. (c) The zero-bias Hall resistivity versus $1/\delta$ for current driving along the $x$-direction and $y$-direction respectively. The $y$-directional zero-bias Hall resistivity decreases with an increasing QML while the $x$-direction response remains quantized.
• Figure 5: Fraunhofer pattern of flat-band QAH Josephson junctions. (a) An illustration of a flat-band QAH Josephson junction with chiral Andreev edge states coupled to each other controlled by the QML, allowing local Andreev reflections in the junction. (b) A $2\Phi_0$-$\Phi_0$ crossover in the Fraunhofer pattern occurs when increasing the QML by reducing $\delta$. The weak link has length $d=100a$ and width $W = 30a$ and is composed of the Lieb-QWZ lattice with $\alpha=0.5$, $\tilde{t} = 0.8t$.