Quantum geometry in condensed matter.

One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of electrons in condensed matter can be characterized by the geometry of the Hilbert space of their wave functions. Such quantum geometry, comprising Berry curvature and the quantum metric, can thus exert profound influences on various properties of materials. The dipoles of both Berry curvature and the quantum metric produce nonlinear transport. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. The uniformly distributed momentum-space quantum geometry stabilizes the fractional Chern insulators and results in the fractional quantum anomalous Hall effect. Here we review in detail quantum geometry in condensed matter, paying close attention to its effects on nonlinear transport, superconductivity and topological properties. Possible future research directions in this field are also envisaged.