Linear displacement current solely driven by the quantum metric

Quantum metric and Berry curvature are the real part and imaginary part of the quantum geometric tensor, respectively. The $\mathcal{T}$-odd ($\mathcal{T}$, time-reversal) nonlinear Hall effect driven by the quantum metric dipole, recently confirmed in [Science 381, 181 (2023)] and [Nature (London) 621, 487 (2023)], established the geometric duality to the $\mathcal{T}$-even nonlinear Hall effect that driven by the Berry curvature dipole. Interestingly, a similar geometric duality between the quantum metric and the Berry curvature, particularly for the linear response of Bloch electrons, has not been established, although the $\mathcal{T}$-odd linear intrinsic anomalous Hall effect (IAHE) solely driven by the Berry curvature has been known for a long time. Herein, we develop the quantum theory for displacement current under an AC electric field. Particularly, we show that the $\mathcal{T}$-even component of the linear displacement current conductivity (LDCC) is solely determined by the quantum metric, by both the response theory and the semiclassical theory. Notably, with symmetry analysis we find that the $\mathcal{T}$-even LDCC can contribute a Hall current in $\mathcal{T}$-invariant systems but with low symmetry, while its longitudinal component is immune to symmetry. Furthermore, employing the model Hamiltonians, we arrive at a $1/\ensuremath{\mu}$ ($\ensuremath{\mu}$, chemical potential) experimental observable enhancement of the displacement current owing to the divergent behavior of quantum metric near Dirac point, similar to the IAHE at Weyl point. Our study reveals the band geometric origin of the linear displacement current and establishes, together with the IAHE, the geometric duality for the linear response of Bloch electrons. Additionally, our paper offers the intrinsic Hall effect in $\mathcal{T}$-invariant materials, which cannot be envisioned in DC transport in both linear and nonlinear regimes.