  Quantum metric and Berry curvature are the real part and imaginary part of
the quantum geometric tensor, respectively. The T-odd (T: time-reversal)
nonlinear Hall effect driven by the quantum metric dipole, recently confirmed
in Science 381, 181 (2023) and Nature 621, 487 (2023), established the
geometric duality to the 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 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 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 T-even
LDCC can contribute a Hall current in T-invariant systems but with low
symmetry, while its longitudinal component is immune to symmetry. Furthermore,
employing the Dirac Hamiltonian, we arrive at a $1/\mu$ ($\mu$: 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 work 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
work offers the very first intrinsic Hall effect in T-invariant materials,
which can not be envisioned in DC transport in both linear and nonlinear
regimes.
