$N$-bein formalism for the parameter space of quantum geometry

TL;DR

This paper introduces the $N$-bein formalism, a geometric framework generalizing the quantum geometric tensor, to analyze quantum state variations, their commutativity, and associated physical observables using Cartan-like geometry.

Contribution

It presents the $N$-bein as a novel geometric object that extends the quantum geometric tensor, enabling new insights into quantum state parameter space and physical invariants.

Findings

The $N$-bein captures state correlations not accessible by previous methods.

The formalism relates torsion to the anti-symmetric part of the quantum geometric tensor.

Applications to harmonic oscillators demonstrate the new tensors' physical relevance.

Abstract

This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it $N$-bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the $N$-bein behaves like a ``square root'' of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the $N$-bein allows us to introduce a notion of torsion and curvature \`{a} la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods.