Hall effect in topologically trivial isolated flat-band systems

Abstract

We study the Hall effect in topologically trivial isolated flat-band systems (i.e., flat bands are separated from other bands and have zero Chern number) for a weak magnetic field. In a naive semiclassical picture, the Hall conductivity vanishes when dispersive bands are unoccupied, since there are no mobile carriers. To go beyond the semiclassical picture, we establish a fully quantum mechanical gauge-invariant formula for the Hall conductivity that can be applied to any lattice models. We apply the formula to a general $N+M$-band model with $N$ dispersive bands and $M$-fold degenerate isolated flat bands, and find that when the dispersive bands are unoccupied, the total conductivity takes a universal form consisting of the energy difference between the dispersive and flat bands, and the non-Abelian quantum geometric tensor of the flat bands, which can be nonzero in systems with vanishing Berry curvature. We numerically confirm the Hall effect for isolated flat-band lattice models on the honeycomb lattice ($N=M=1$) and two different Kagome lattices ($N=2$, $M=1$ and $N=1$, $M=2$).

Figures (5)

• Figure 1: (a) Gapped honeycomb lattice model of nearest-neighbor and next-nearest-neighbor hoppings (black, red-dashed, and blue-dotted lines correspond to the hopping strength $\sqrt{3}t$, $t$, and $0$), having one isolated flat band at $E=0$Misumi2017. (b) Gapped Kagome lattice model with nearest-neighbor hoppings of different strengths (black, red, and blue lines correspond to the hopping strengths $t$, $\alpha t$, $t/\alpha$; $\alpha=2$), having one isolated flat band at $E=0$Bilitewski2018. (c) Kagome-$3$ lattice model with nearest, next-nearest, and next-next-nearest neighbor hoppings with the same amplitude $t$, having two-fold degenerate isolated flat bands at $E=0$Bergman2008. [(d),(e),(f)] Band structures of the models of (a), (b), and (c), respectively. Energy is measured in units of $t$.
• Figure 2: The normalized total Hall conductivity $(\sigma_{xy}^{\text{Total}}/\sigma_{xy0})\cdot(\Gamma/\Lambda)$ of the gapped honeycomb ($\Lambda\approx0.184$, red), gapped Kagome ($\Lambda\approx0.036$, blue), and Kagome-$3$ model ($\Lambda\approx6.357$, green). Eq. (\ref{['Around_flat_bands']}) gives the solid lines for $\Gamma=0.2$, $0.1$, and $0.05$ from outside to inside. The markers are calculated by Eqs. (\ref{['eq:TR']}) and (\ref{['eq:TH']}) for $\Gamma=0.2$ (circle), $0.1$ (triangle), and $0.05$ (diamond). A finite small value at $\mu=-0.5$ is subtracted in the markers.
• Figure 3: (a) The modified band structure of the gapped honeycomb lattice model with $\delta t=0.2$. (b) $\Gamma$-dependence of the maximum absolute value of the total Hall conductivity $|\sigma_{xy}^{\text{Total}}/\sigma_{xy0}|$ around the isolated flat band ($\mu\in[-t,t]$). The red circle, blue triangle, and green diamond curves correspond to the case of $\delta t=0.5$, $0.2$, and $0$, respectively.
• Figure 4: Plots of the geometric part of the formula of the models in the main text. The dashed lines represent the boundary of the first Brillouin zone. (a) Honeycomb lattice model. (b) Gapped Kagome lattice model for the lower dispersive band ($n=1$). (c) Gapped Kagome lattice model for the upper dispersive band ($n=2$). (d) Kagome-$3$ lattice model.
• Figure 5: Transport ($\sigma_{xy}^{\text{TR}}$, blue) and thermodynamic ($\sigma_{xy}^{\text{TH}}$, red) contributions to the Hall conductivity for the honeycomb lattice model with $m=3.0$, $T=0$, $\delta t=0.2$, and $\Gamma=0.1$.