Tunable quantum metric and band topology in bilayer Dirac models

TL;DR

The paper addresses how to achieve simultaneous tunability of quantum metric and band topology in condensed-matter systems by introducing a bilayer Dirac model formed from two Dirac Hamiltonians with energy scales ε1 ≫ ε2 and a weak interlayer coupling λ. By inducing a band inversion in the layer subspace, the authors show tunable topology across all AZ symmetry classes and a quantum metric that scales as g ∝ 1/λ^2 near the inversion point, leading to enhanced quantum weight. They analyze three pathways to realize topological flat bands and demonstrate that the quantum metric can significantly affect boundary-state localization, introducing a geometry-induced length scale ξg that governs decay in the nearly flat-band regime. The framework provides a versatile platform for exploring the interplay between band topology and quantum geometry, with potential realizations in moiré materials and metamaterials that enable experimental studies of geometry-driven edge physics.

Abstract

Quantum metric, a fundamental component of quantum geometry, has attracted broad interest in recent years due to its critical role in various quantum phenomena. Meanwhile, band topology, which serves as an important framework in condensed matter physics, has led to the discovery of various topological phases. In this work, we introduce a bilayer Dirac model that allows precise tuning of both properties. Our approach combines two Dirac Hamiltonians with distinct energy scales; one producing relatively dispersive bands and the other yielding relatively flat bands. The dispersive and flat bands are weakly coupled via hybridization $λ$. By inducing a band inversion in the layer subspace, we achieve flexible tuning of band topology across all Altland-Zirnbauer symmetry classes and quantum metric scaling as $g \propto 1/λ^2$ near band inversion point. Using the bilayer Su-Schrieffer-Heeger model, we investigate the localization properties of gapless boundary states, which are affected by quantum metric. Our work lays a foundation for exploring the interplay between band topology and quantum metric.

Figures (5)

• Figure 1: (a) The relatively dispersive (green) and flat (red) bands. (b) The illustration of band inversion between the dispersive and flat bands at the $M$ point in the “layer” subspace. The colorbar encodes the expectation value of the “layer” pseudospin $\tau_z$. For the plot, the common model parameters are taken as $\epsilon_1=2$, $\epsilon_2=-0.4$, $m_1=0.2$, $m_2=0.6$ for the “bilayer” SSH model. In (b), $\lambda=0.2$ is taken.
• Figure 2: Three distinct cases for realizing topological flat bands. The colorbar encodes the expectation value of inversion operator $I$. The band topology are characterized by the topological invariant $\nu$. The same model parameters are used for (a)-(c) as those used in Fig. \ref{['fig1']}. In (d)-(f), we take $\epsilon_1=2$, $\epsilon_2=0.2$, $m_1=-0.2$, and $m_2=0.6$ for the plot. In (c)-(g), we take $\epsilon_1=-2$, $\epsilon_2=0.2$, $m_1=-0.2$, and $m_2=0.6$ for the plot.
• Figure 3: (a) The distribution of the quantum metric $g_{xx}(k_x)$ for the “bilayer” SSH model. (b) The quantum weight as a function of $\lambda$ under different settings of $\epsilon_1$. (c) The distribution of the quantum metric $g_{xx}(k_x,k_y)$ for the “bilayer” Qi-Wu-Zhang model. (d) The quantum weight of the dispersive (flat) bands and their fitting results for the “bilayer” Qi-Wu-Zhang model as a function of $\lambda$, with $c_0=0.0001$ and $c_1=5$. The common model parameters are taken as $\epsilon_2=-0.4$, $m_2=0.6$, and $m_1=0.2$. In (a), we take $\epsilon_1=20$ and $\lambda=0.2$. In (c), we take $\epsilon_1=20$ and $\lambda=0.02$. In (d), we take $\epsilon_1=20$.
• Figure 4: (a) The plot of $\text{ln}|\psi(x)|$ for end states in the two-band SSH model. (b) The plot of $\text{ln}|\psi(x)|$ for end states in the “bilayer” SSH model. (c) The quantum weight $G_{xx}$ as a function of $\epsilon_1$ for the “bilayer” SSH model. (d) The plot of $\text{ln}|\psi(x)|$ for end states in the “bilayer” SSH model under different setting of $\epsilon_1$. The common model parameters are taken as $m_2=0.6$, $\epsilon_2=-0.4,\lambda=m_1=0.2$. In (b), we take $\epsilon_1=15$.
• Figure 5: Quantum weight and edge states plot for the “Bilayer” SSH model. (a) The plot of quantum weight $G_{xx}$ as a function of $m_1$. (b) The plot of $\text{ln}|\psi(x)|$ for the end states under different setting of $m_1$. (c) The plot of quantum weight $G_{xx}$ as a function of $\lambda$. (d) The plot of $\text{ln}|\psi(x)|$ for the end states under different setting of $\lambda$. In (a) and (b), $m_2=0.6$, $\epsilon_1=20$, $\lambda=0.2$, and $\epsilon_2=-0.4$. In (c) and (d), $m_2=0.6$, $\epsilon_1=15$, $\epsilon_2=-0.4$, and $m_1=0.2$.