Beyond the Berry Phase: Extrinsic Geometry of Quantum States

TL;DR

This paper introduces a scalar gauge-invariant invariant, the Bargmann invariant, which fully characterizes the geometric properties of quantum states and reveals new extrinsic geometric features beyond the Berry phase.

Contribution

It demonstrates how the Bargmann invariant encodes all geometric information of quantum states, including extrinsic properties, extending beyond traditional Berry phase and curvature concepts.

Findings

Bargmann invariant fully describes quantum state geometry.

Higher order terms reveal extrinsic geometric properties.

Applications to polarization, electrical response, and flat band physics.

Abstract

Consider a set of quantum states $| \psi(x) \rangle$ 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 $P^{(3)}(x_1, x_2, x_3)=\operatorname{tr}[P(x_1) P(x_2)P(x_3)]$, where $P(x) = |\psi(x)\rangle \langle\psi(x)|$. 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 determined 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 $\omega$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $\omega$ 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 electrical response to a modulated field and physics of flat bands.