Extrinsic geometry of quantum states

Consider a set of quantum states |ψ(x)) parameterized by x taken from some parameter space M. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant P(3)(x1,x2,x3)=tr[P(x1)P(x2)P(x3)], where P(x)=|ψ(x))(ψ(x)| are the projectors. Mathematically, P(x) defines a map from M to the complex projective space CPn and this map is uniquely determined by P(3)(x1,x2,x3) up to a symmetry transformation. The phase argP(3)(x1,x2,x3) can be used to compute the Berry phase for any closed loop in M, however, as we prove, it contains other information that cannot be extracted from any Berry phase. When the arguments xi of P(3)(x1,x2,x3) are taken close to each other, to the leading order, it reduces to the familiar Berry curvature ω and quantum metric g. We show that higher orders in this expansion are functionally independent of ω and g and are related to the extrinsic properties of the map of M into CPn giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor T. Finally, we show how our results have immediate applications to the modern theory of polarization, calculation of the electric current response to a modulated field and physics of flat bands.