Quantum Metric Length as a Fundamental Length Scale in Disordered Flat Band Materials

Abstract

Our previous understanding of electronic transport in disordered systems was based on the assumption that there is a finite Fermi velocity for the relevant electrons. The Fermi velocity determines important length scales in disordered systems such as the diffusion length and the localization length. However, in disordered systems with vanishing or nearly vanishing Fermi velocity, it is uncertain what determines the important length scales in such systems. In this work, we use the 1D Lieb lattice with isolated flat bands as an example to show that the quantum metric length (QML) is a fundamental length scale in the ballistic, diffusive and localization regimes. The QML is defined through the Bloch state wave functions of the flat bands. In the ballistic regime with short junctions, the QML controls the finite energy transport properties. In the localization regime with long junctions, the localization length is determined by the QML and remarkably, independent of disorder strength over a wide range of disorder strength. We call this unconventional localization regime, the quantum metric localization regime. In the diffusive regime, we demonstrate that the diffusion coefficient is linearly proportional to the QML via the wave-packet dynamics numerically. Importantly, the numerical results are consistent with the analytical results obtained through the Bethe-Salpeter equation. We conclude that the QML is a fundamentally important length scale governing the properties of disordered flat band materials.

Figures (3)

• Figure 1: (a) A 1D Lieb lattice, which contains three sub-lattices (A, B, and C, respectively) per unit cell. (b) A schematic band structure of a 1D Lieb lattice with an isolated flat band (red in color). (c) A schematic M/FB/M junction used to study the transport property of a Lieb lattice with disorder. The external leads are connected to the Lieb lattice with coupling strength $T_\partial$, and $V$ denotes the voltage difference of the two leads.
• Figure 2: (a) The $|\psi(x)|^{2}=\sum_{\alpha=ABC}|\psi_{\alpha}(x)|^{2}$ of an interface state at the M/FB/M junction where $\psi_{\alpha}(x)$ is the $\alpha$ sub-lattice component of the wave function. Within the Lieb lattice, the probability density of $\psi_{\alpha}(x)$ decays exponentially away from the M/FB interface with a length scale $\lambda=4\ell_\mathrm{QM}$. (b) Transmission probability $\mathcal{T}_{LR}$ for a $L=100$ junction without disorder with the setup of the lower panel of (a). No zero energy transmission is observed, and the transmission at finite energy is mediated by interface states with energy $E_0$. (c) Zero energy transmission arises when disorder is introduced. (d) The transmission $\mathcal{T}$ at zero energy $E=0$ for varying junction length $L$ when $\Gamma/E_0=300$ with the setup illustrated in Fig. \ref{['fig:1']}(c). Three distinct transport regimes are observed where the shaded region indicates the diffusive regime. The diffusive regime is characterized by the $1/L$ scaling, as depicted by the red dashed line. (e) The localization length $\xi$ of the localization regime for varying disorder strength $\Gamma$ for three sets of QML $\ell_\mathrm{QM}$. $\xi$ increases in the weak disorder limit, reaching a plateau of $\xi\sim 4\ell_\mathrm{QM}$ when disorder strength is comparable with the band gap $\Delta$. The parameters in the calculations are chosen as $t_{N}=1$, $J=1000$, $\delta=0.01$, $T_{\partial}=0.1$, $E_{0}(\delta)=4T_{\partial}^{2}\delta/t_N$ for (b-e).
• Figure 3: (a) The time evolution of the site occupation $\langle n_x(t)\rangle=\langle\sum_\alpha |\psi_{\alpha}(x,t)|^2\rangle$ for the wave packet $\psi(t)$, where $\psi(t=0)$ is localized and projected to the flat band states. (b) The MSD $\Delta X^2(t)$ in (a). The simulation is performed with $\Gamma/\Delta=0.05$. It is clear that $\Delta X^{2}(t)\propto 2Dt$ which indicates a diffusive behavior. (c) The linear fit of $D$ as function of the QML $\ell_{\rm{QM}}$ as predicted from Eq. \ref{['eq:diffusion']}, for disorder strengths $\Gamma/\Delta=0.05$ and $\Gamma/\Delta=0.1$ respectively. The simulation is performed on a 1D Lieb lattice with length $L=401$ in (a-b) and $L=1001$ in (c). All results are obtained by averaging over $500$ disorder configurations.