N-bein Formalism in Quantum Geometry

• N-bein formalism is a geometric framework that provides a square root decomposition of the quantum metric, analogous to the vielbein in Riemannian geometry.
• It encapsulates the quantum geometric tensor, decomposing it into quantum metric and Berry curvature components while introducing quantum torsion to capture order-sensitive parameter variations.
• The framework yields gauge-invariant observables and has practical applications in analyzing harmonic and generalized oscillators to expose refined quantum state correlations.

The $N$-bein formalism is a geometric framework introduced to generalize the quantum geometric tensor (QGT) structure over the parameter space of quantum systems. Drawing an analogy with the vielbein (orthonormal frame) in Cartan's formalism of Riemannian geometry, the $N$-bein provides a "square root" decomposition of the quantum metric and Berry curvature, and enables a refined description of quantum correlations, torsion, and curvature in parameter space (Romero et al., 2024).

1. Definition and Structure of the $N$-bein

Let $\{\lambda^{\mu}\}_{\mu=1}^N$ be a set of real parameters on which a family of normalized eigenstates $|n(\lambda)\rangle$ depend. The $N$-bein, denoted $E^{I}{}_{\mu}(\lambda)$, defines an orthonormal frame on parameter space analogously to a non-adiabatic coupling vector.

The quantum geometric tensor (QGT) for a given eigenstate $|n\rangle$ is

$$g_{\mu\nu}^{(n)}- \tfrac{i}{2}F_{\mu\nu}^{(n)} =\langle\partial_{\mu} n|\,\bigl(1-|n\rangle\langle n|\bigr)\,\partial_{\nu} n\rangle,$$

which decomposes via the $N$-bein as

$N$0

The $N$1-bein thus satisfies

$N$2

where $N$3 is the real part (the quantum metric) and $N$4 is the Berry curvature. The $N$5-bein is interpreted as a "square root" of the metric.

2. Two-State Geometric Tensor and Quantum Torsion

To extend the geometric description to probe correlations between two distinct quantum levels $N$6 and $N$7, the two-state geometric tensor is introduced:

$N$8

which, in terms of $N$9-beins for each state, reads

$N$0

The symmetric and anti-symmetric parts are defined as

$N$1

$N$2

where $N$3 is a complex metric-like object and $N$4 is interpreted as a quantum torsion: $N$5 A vanishing torsion indicates commutativity of consecutive parameter variations $N$6; nonzero torsion marks the presence of non-trivial quantum-state correlations sensitive to the order of parameter changes.

3. Connections, Cartan Structure Equations, and Differential Forms

Under local gauge transformations $N$7, the usual Berry connection $N$8 transforms by a shift, while $N$9 acquires a relative phase.

A new Abelian connection is introduced,

$\{\lambda^{\mu}\}_{\mu=1}^N$0

which is distinct from either individual Berry connection. In the language of differential forms, the $\{\lambda^{\mu}\}_{\mu=1}^N$1-bein and connection are written as

$\{\lambda^{\mu}\}_{\mu=1}^N$2

The torsion $\{\lambda^{\mu}\}_{\mu=1}^N$3-form and curvature $\{\lambda^{\mu}\}_{\mu=1}^N$4-form become

$\{\lambda^{\mu}\}_{\mu=1}^N$5

$\{\lambda^{\mu}\}_{\mu=1}^N$6

where $\{\lambda^{\mu}\}_{\mu=1}^N$7 is the Berry curvature $\{\lambda^{\mu}\}_{\mu=1}^N$8-form for level $\{\lambda^{\mu}\}_{\mu=1}^N$9.

The Cartan structure equations are satisfied: $|n(\lambda)\rangle$0 with associated Bianchi identities,

$|n(\lambda)\rangle$1

4. Component Expressions: Torsion and Curvature

Let $|n(\lambda)\rangle$2 denote the inverse $|n(\lambda)\rangle$3-bein, so $|n(\lambda)\rangle$4. The torsion tensor in mixed indices reads

$|n(\lambda)\rangle$5

The Riemann-like curvature is given by

$|n(\lambda)\rangle$6

5. Gauge-Invariant Observables and Physical Interpretation

Although $|n(\lambda)\rangle$7, $|n(\lambda)\rangle$8, and $|n(\lambda)\rangle$9 are not fully gauge-invariant, one can form gauge-invariant physical observables using $N$0-invariant combinations of the real and imaginary parts. For any complex two-form $N$1, the invariants include:

Invariant Type	Expression	Description
Norm-type	$N$2	Quadratic, norm-like observable
Area-type	$N$3	Area-like invariant
Scalar	$N$4	Contracted scalar invariant using the quantum metric

All are invariant under local gauge transformations $N$5, with $N$6 the inverse quantum metric. These quantities generate new physical observables and provide additional probes of quantum state correlations.

6. Examples: Harmonic and Generalized Oscillators

Two explicit models illustrate the structure and physical content of the $N$7-bein formalism:

(a) Harmonic oscillator with linear term.

Hamiltonian:

$N$8

Eigenstates are displaced Hermite functions. Exactly four nonzero $N$9-beins are present, purely imaginary and independent of $E^{I}{}_{\mu}(\lambda)$0. The quantum geometric tensor is diagonal:

$E^{I}{}_{\mu}(\lambda)$1

with scalar curvature

$E^{I}{}_{\mu}(\lambda)$2

Two-state torsion $E^{I}{}_{\mu}(\lambda)$3 is nonzero only for $E^{I}{}_{\mu}(\lambda)$4, showing that the order of $E^{I}{}_{\mu}(\lambda)$5 variations only matters between adjacent levels. Scalar invariants $E^{I}{}_{\mu}(\lambda)$6 peak at $E^{I}{}_{\mu}(\lambda)$7 and decay rapidly with $E^{I}{}_{\mu}(\lambda)$8.

(b) Generalized oscillator with linear term.

Hamiltonian:

$E^{I}{}_{\mu}(\lambda)$9

All four $|n\rangle$0-beins become fully complex and $|n\rangle$1-dependent. The metric $|n\rangle$2 and Berry curvature $|n\rangle$3 acquire $|n\rangle$4-sensitive contributions through $|n\rangle$5, while $|n\rangle$6 delivers the oscillator contribution. Torsion remains nonzero only for $|n\rangle$7. Scalar invariants $|n\rangle$8 fall with $|n\rangle$9, indicating strong state-to-state torsional effects primarily for low-lying states.

In both models, the $$g_{\mu\nu}^{(n)}- \tfrac{i}{2}F_{\mu\nu}^{(n)} =\langle\partial_{\mu} n|\,\bigl(1-|n\rangle\langle n|\bigr)\,\partial_{\nu} n\rangle,$$0-bein framework exposes how parameter variations in one state are mediated through intermediate quantum levels, and the antisymmetric component of the two-state quantum geometric tensor (torsion) quantifies previously inaccessible quantum-state correlations in parameter space.

7. Significance and Scope

The $$g_{\mu\nu}^{(n)}- \tfrac{i}{2}F_{\mu\nu}^{(n)} =\langle\partial_{\mu} n|\,\bigl(1-|n\rangle\langle n|\bigr)\,\partial_{\nu} n\rangle,$$1-bein formalism establishes a precise geometric structure on parameter spaces of quantum systems, generalizing familiar concepts such as frames, connections, torsion, and curvature from differential geometry to quantum state manifolds. The introduction of torsion and new gauge-invariant observables enables characterization of the order dependence of parameter variations and the emergence of nontrivial quantum correlations that are not captured by traditional metrics or Berry curvature alone. This approach provides a systematic platform for investigating quantum state geometry, its associated topological and geometric phases, and their observable consequences in both simple and highly structured quantum systems (Romero et al., 2024).