Asymmetric quantum geometry: Third cumulant tensor and inversion breaking

Abstract

The quantum geometric tensor characterizes the local geometry of quantum states in parameter space, with its real part given by the quantum metric and its imaginary part proportional to the Berry curvature. As the second cumulant of the position operator, it motivates the study of higher-order cumulants, which are generally complex and accessible via a generating function. In analogy with the third cumulant in classical probability, the third cumulant tensor (TCT) is expected to capture asymmetry-related properties, while also encoding phase-sensitive information. Motivated by this, we investigate the TCT in the one-dimensional Rice–Mele model under controlled breaking of onsite- and bond-inversion symmetries. We find that breaking at least one inversion symmetry yields a nontrivial quantum metric and imaginary TCT, whereas a nonzero real TCT arises only when both symmetries are broken. These results establish a clear connection between the TCT and spatial inversion symmetry and motivate further studies of its role in nonlinear effects.