Quantum Geometry

The quantum geometric tensor takes the form,

$$F_{\mu \nu }^{(n)}=⟨{\check{\partial }}_{\mu }{u}_{n,\mathbf{k}}|{\check{\partial }}_{\nu }{u}_{n,\mathbf{k}}⟩$$

It can be decomposed into a real and imaginary part. The Quantum metric $g$ describes distance between wavefunctions, and is the real part

$$F=g+\frac{i}{2}\Omega$$

The Berry curvature is the imaginary component.

Quantum metric nonlinear Hall effect

PT-symmetric AFM without rotational $\Omega =0$, $M=0$ $AHE=0$ $g$ is nontrivial

• Apply $E$ perpendicular to magnetic dipole directions, ${\Omega }_E={\nabla }_k\times (gE)$
• Apply another $E$, $V_{NHE}\propto E\times {\Omega }_E=E\times ({\nabla }_k\times (gE))$