Many faces of quantum geometric tensor in non-Hermitian systems

Abstract

The interplay between the quantum geometric tensor (QGT) and non-Hermitian quantum mechanics is essential for characterizing both the geometric structure and dissipative dynamics of quantum systems. In non-Hermitian settings, however, the non-uniqueness of inner products in Hilbert space leads to multiple inequivalent definitions of the QGT. In this work, we systematically study these distinct formulations using the modified Bender Hamiltonian. We show that the QGT defined within both biorthogonal and metricized frameworks diverges as the exceptional point is approached. Despite this shared divergence, the two formulations yield different quantum metrics and Berry curvatures, defined as the symmetric and antisymmetric components of the QGT, respectively. In contrast, the QGT constructed solely from right eigenstates, analogous to the Hermitian case, fails to capture the divergence of the quantum metric at the exceptional point.