Quantum-geometry-enabled Landau-Zener tunneling in singular flat bands

TL;DR

This work addresses whether singular flat bands can support dc transport under a static electric field and reveals that quantum geometry governs the response. Using a minimal two-band lattice with tunable maximal quantum distance $d$, the authors derive the Wannier-Stark spectrum and identify two geometric phases: a Berry-phase–driven intraband phase away from the singular points, and two far-reaching phases near the singular band-crossing point that control Landau-Zener tunneling via $\theta$ (tunneling rate) and $\varphi$ (a generalized Berry phase). Away from the crossing points, the flat-band states remain exponentially localized with transport suppressed; near the crossing, interband coupling bends the Wannier-Stark ladder and delocalizes the wavefunctions, enabling nontrivial flat-band transport that is fully characterized by $(\theta,\varphi)$ and governed by the single parameter $d$. The framework is validated in a kagome-lattice model, showing the results extend to realistic SFB systems and highlighting a quantum-geometric pathway to transport phenomena in flat-band materials.

Abstract

Flat-band materials have attracted substantial interest for their intriguing quantum geometric effects. Here we investigate how singular flat bands (SFBs) respond to a static, uniform electric field and whether they can support single-particle dc transport. By constructing a minimal two-band lattice model, we show that away from the singular band crossing point (BCP), the Wannier-Stark (WS) spectrum of the flat band is well captured by an intraband Berry phase $Φ_{\mathrm{B}}$. The associated WS eigenstates are exponentially localized along the field direction, precluding dc transport. In contrast, near the BCP the interband Berry connection becomes prominent and drives Landau-Zener tunneling, which bends the flat-band WS ladder and delocalizes the SFB wavefunctions. Remarkably, this regime is governed solely by the maximal quantum distance $d$ through two geometric phases $(θ,\varphi)$: $θ$ characterizes the tunneling rate and $\varphi$ acts as a generalized Berry phase. These results highlight the essential role of quantum geometry in enabling nontrivial transport signatures in SFBs.

Figures (5)

• Figure 1: (a) Band dispersion showing a flat band (blue) tangent to a dispersive band (orange) at $\Gamma$$(0,0)$ and X $(\pi,0)$, with chirality (valley index) $\xi = \pm 1$, respectively. The band structure is independent of $d \in [0,1]$. (b) Compact localized state (CLS) of the flat band. Concentric blue and black dots denote two orbitals on each lattice site. Colored numbers indicate the wavefunction amplitudes of the CLS, which consists only of the orbitals shown in the corresponding dark color. Gray dashed lines mark unit cells. (c) Wannier charge center flow of the flat band versus $k_x$ for $d=0.8$. As $k_x$ crosses a singular BCP, the Berry phase jumps by $\xi \Delta \Phi_{\mathrm{B}}$. (d) Real-space WS wavefunction of the flat band versus $y$ (semi-log scale) at $k_x=0.1\pi$ for $d=1$, $0.8$, and $0.2$.
• Figure 2: (a--d) Numerical WS spectra (in units of $\Delta = e F a$) for $F/t = 0.002$ applied along $y$, plotted versus $k_x$ for several values of $d$. Orange dashed lines in (a--c) guide the eye for the flat-band WS levels and follow the dispersion set by $\varphi$. (e--g) Corresponding geometric phases $\theta$ (blue) and $\varphi$ (orange) for (a--c). (h) Band-resolved weights of the WS eigenstates for $d=1$.
• Figure 3: (a) Upper: kagome lattice with $x$ and $y$ directions indicated. Solid and dashed blue boxes denote the primitive unit cell and the enlarged unit cell, respectively. Lower: BZ and high-symmetry points; gray lines indicate the $k$-space periodicities $2k_{0x}$ and $2k_{0y}$. (b),(c) WS spectra for $F / t = 0.01$ applied along $y$ and $x$, plotted versus $k_x$ and $k_y$ in units of $\Delta = \pi e F / k_{0y}$ and $\Delta = \pi e F / k_{0x}$, respectively. Gray lines: lattice calculation; blue lines: prediction from the geometric phases $(\theta,\varphi)$ for $d=1$.
• Figure S1: (a) Compact localized state (CLS) of the flat band. Concentric blue and black dots denote two orbitals on each lattice site. Colored numbers indicate the wavefunction amplitudes of the CLS, which consists only of the orbitals shown in the corresponding dark color. Gray dashed lines mark unit cells. (b) A representative interference channel demonstrating perfect destructive interference. The arrows indicate hoppings to a $B$ orbital outside the CLS; the numbers and colors denote the hopping amplitudes and their originating orbitals, respectively.
• Figure S2: (a) Two representative real-space wavefunctions with adjacent energies, plotted as a function of $y$ on a semi-log scale, for $d=1$, $F/t=0.002$, and $k_x=0.1\pi$. (b) Same as (a) but with $k_x=0.03\pi$. (c) Same as (b) but with $F/t=0.004$. (d) Band-resolved weights $|\phi_{0,1}|^2$ as a function of $k_y$, corresponding to the lower panel of (b). (e) Phase difference $\Delta\mathrm{Arg}(\phi_{0,1})$ between the exact numerical solution and the analytical prediction in Eq. \ref{['eq:phi01']}, corresponding to (d).