Disorder-Induced Singularity of the Quantum Metric

The quantum weight is a new concept to describe gap electronic states of matter. This quantity is obtained by integrating the quantum metric (the real part of the quantum metric tensor) in the same way as the Berry phase is obtained by integrating the Berry curvature (the imaginary part of the quantum metric tensor). The quantum weight determines a number of kinetic quantities such as the nonlinear anomalous Hall effect, optical conductivity, and photovoltaic effect. In this work, it is shown that nonmagnetic disorder in topological insulators can induce a singularity in the quantum metric and quantum weight.