Extrinsic geometry of quantum states

Abstract

Consider a set of quantum states $|\ensuremath{\psi}(x)\ensuremath{\rangle}$ 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 ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})=tr[P({x}_{1})P({x}_{2})P({x}_{3})]$, where $P(x)=|\ensuremath{\psi}(x)\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}(x)|$ are the projectors. Mathematically, $P(x)$ defines a map from $M$ to the complex projective space $\mathbb{C}{P}^{n}$ and this map is uniquely determined by ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ up to a symmetry transformation. The phase $arg{P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ 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 ${x}_{i}$ of ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ are taken close to each other, to the leading order, it reduces to the familiar Berry curvature $\ensuremath{\omega}$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $\ensuremath{\omega}$ and $g$ and are related to the extrinsic properties of the map of $M$ into $\mathbb{C}{P}^{n}$ 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.

Topics & Concepts