Scaling of quantum geometry near non-Hermitian topological phase transitions

The geometry of quantum states can be an indicator of criticality, yet it remains less explored under non-Hermitian topological conditions. In this work, we unveil diverse scalings of the quantum geometry over the ground-state manifold close to different topological phase transitions in a non-Hermitian long-range extension of the Kitaev chain. The derivative of the geometric phase, as well as its scaling behavior, shows that systems with different long-range couplings can belong to a distinct critical class. Near certain criticalities, we further find that the Wannier state correlation function associated with the extended Berry connection of the ground state exhibits spatially anomalous behaviors. Finally, we analyze the scaling of the quantum geometric tensor near phase transitions across exceptional points, shedding light on the emergence of a distinct critical class.