Disorder-induced diffusion transport in flat-band systems with quantum metric

TL;DR

The paper addresses how disorder can induce diffusive transport in flat-band systems where the Fermi velocity vanishes. By combining a diamond-lattice M/FB/M junction, wave-packet dynamics, and a diagrammatic framework (self-consistent Born approximation and Bethe-Salpeter/ diffuson treatment), it shows that disorder activates bulk diffusion with a diffusion coefficient D that scales as D ≈ C\Gamma\overline{\mathcal{G}} (with C ≈ 0.337) and that the diffusion length is set by the Brillouin-zone-averaged quantum metric \overline{\mathcal{G}}. Importantly, disorder generates an effective velocity operator, yielding finite DC transport and revealing a disorder-driven delocalization mechanism not captured by conventional quantum-diffusion theories. The results, relevant to moiré and flat-band materials such as twisted bilayer graphene, provide a concrete mechanism by which quantum geometry governs diffusion in the presence of disorder, with practical implications for designing transport in flat-band devices.

Abstract

Our previous understanding of transport in disordered system depends on the assumption that there is a well-defined Fermi velocity. The Fermi velocity determines important length scales in the system such as the diffusion length and localization length. However, nearly flat band materials with vanishing Fermi velocity, it is uncertain how to understand the disorder effects and what quantities determine the characteristic length scales in the system. In the clean limit, it is expected that the bulk transport is absent. In this work, we demonstrate, with a diamond lattice, that disorder can induce diffusion transport in a flat-band system with finite quantum metric. As disorder increases, the bulk transmission channels are activated, and the conductance reaches a maximum before decays inversely with disorder strength. Importantly, via the calculation of the wave-packet dynamics numerically, we show that the quantum metric determines the diffusion length of the system. Analytically, we show that the interplay between the disorder and quantum geometry gives rise to an effective Fermi velocity, as captured by the self-consistent Born approximation. The diffusion coefficient is identified from the Bethe-Salpeter equation under the ladder approximation. Our results reveal a disorder-driven delocalization mechanism in flat-band systems with finite quantum metric which cannot be understood by well-established theories of quantum diffusion. Our theory is important for understanding the disorder effects and transport properties of flat band materials such as twisted bilayer graphene which are current under intense investigation.

Figures (11)

• Figure 1: (a) Schematic of the 1D diamond lattice, which contains three sites A, B, C per unit cell. (b) Energy spectrum of the 1D diamond lattice. The central gap is exaggerated for clarity. (c) Schematic of the four-terminal M/FB/M junction. The central disordered 1D diamond chain (blue) of length $L$ serves as the device under measurement, with four metallic leads attached. Lead 1 and 4 are connected to the two ends of the chain, while lead 2 and 3 divide the chain into three segments, forming a $\pi$-shaped configuration. The total length of the disordered part is $L = L_{12} + L_{23} + L_{34}$, with $L_{12} = L_{34} = 10$ fixed throughout this work. Subscripts denote the corresponding lead labels as shown in (c).
• Figure 2: (a) Transmission profiles for varying disorder strength $\Gamma$ at $\delta=0.01$ for a $L=50$ junction. As we increase the disorder strength, transport from bulk states is gradually activated. No zero energy transmission is observed in the clean limit. (b) The transmission $\mathcal{T_0}$ at zero energy $E=0$ and $\mathcal{T}\sim1/L$ fit (red line) for varying junction length $L$ when $\delta=0.01$ and $\Gamma=300E_0$. The gray shaded region indicates the diffusive $1/L$ region. The $\xi$ is chosen as the length when diffusive behavior holds. (c) The transmissions $\mathcal{T}(E=0)$ for different disorder strength at $\delta=0.01$. Transmission contributed from bound states dominates when $\Gamma/E_{0}(\delta)$ is small, and $\mathcal{T}(E=0)$ increases as $\propto\Gamma^{2}$. A further increase in disorder strength enhances transport, peaking at $\Gamma/E_{0}(\delta)\sim200$. (d) The zero energy transmission $\mathcal{T}(E=0)$ for different $\overline{\mathcal{G}}$ at clean limit and a fixed disorder strength $\Gamma=300E_0(\delta=0.01)=0.12$ with $L=50$.
• Figure 3: (a) The time evolution of the site occupation $\langle n_i(t)\rangle=\langle\sum_\alpha |\psi_{i\alpha}|^2\rangle$ for the wave packet $\lvert\psi(t)\rangle$ formed by flat band states. (b) The linear fit of the MSD $\Delta X^2(t)=2Dt$ with $D\approx 0.4109$. The fitted diffusion coefficient $D$ is close to the one value $0.4213$ in Table. \ref{['tab:tab1']} predicted from Eq. \ref{['eq:diffusion']}. The parabolic behavior at the beginning part may contribute to the disorder-free region around the initial wave packet such that it can propagate ballistically shortly. The evolution is performed on 1D diamond lattice with the length $L=401$ under the parameters $\Gamma=0.1, \delta=0.01$ by averaging over $500$ disorder realizations.
• Figure S1: (a) Structure of M/FB/M junction. The flat band originates from the diamond lattice, which has three lattice sites $A,B,C$ per unit cell. (b) Dispersion spectrum of the diamond lattice for $\delta=0.1$. (c) Density distribution of a pair of interface states within the diamond lattice of M/FB/M junction with parameter $\delta=0.01$ and length $L=50$. The two interface states are located at the B- and C- sublattice sites, respectively, with a localization length $\xi=1/(2\delta)$.
• Figure S2: Two-terminal measurement on transmission in the clean limit: (a) transmission profile for different value of $\delta$, while keeping $L\delta\sim1$ and (b) Maximal transmittance and peak energy as a function of the length of the lattice. We note the maximal transmittance is identical, with $E_{0}(\delta=0.04)\sim4E_{0}(\delta=0.01)$. In the inset, we compare the case of having and not having a flat band (2-band model with identical dispersive bands). We note the transmission is highly suppressed when the flat band is removed. In (b), when the junction is long enough, the peak energy $E_p(L)$ approaches a constant value $\sim E_{0}(\delta)$, and the maximal transmittance decays exponentially. In general, the maximal transmittance obeys $\mathcal{T}\sim\mathrm{sech}^{2}(2L\delta)$.