$\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry

AI summary

Overview

Research area: Theoretical quantum physics — quantum geometry of parameter spaces, Cartan/vielbein formalism, non-Abelian (Wilczek–Zee) gauge structures.

Technical level: Advanced. The paper is written for readers comfortable with differential geometry, gauge theory, and the quantum geometric tensor (QGT).

Scope: This paper extends the authors' earlier "𝒩-bein" (N-bein) reformulation of the quantum geometric tensor to quantum systems whose energy levels are degenerate, defining a rectangular degenerate N-bein, a two-state QGT, a torsion-like tensor, gauge-invariant observables, a differential-form version of the formalism, and a worked example of coupled harmonic oscillators in an electric field.

What This Paper Is About

The quantum geometric tensor captures how a quantum state changes when parameters in its Hamiltonian are varied, but its standard construction assumes non-degenerate energy levels. The authors previously introduced the 𝒩-bein — an object analogous to Cartan's orthonormal frame (vielbein) and acting as the "square root" of the QGT — for non-degenerate systems. Here they rebuild that framework for degenerate spectra, where adiabatic transport becomes matrix-valued (Wilczek–Zee) rather than a scalar phase, and where transitions can occur between subspaces of different degeneracy.

Key Contributions

1. A degenerate 𝒩-bein that is a rectangular matrix. The object is defined as e^{(m,n)}_{i;MN} := i⟨m_M|∂_i n_N⟩, with an equivalent Hamiltonian-based form i⟨m_M|∂_i Ĥ|n_N⟩/(E_n − E_m). Because the subspaces ℋ_m and ℋ_n can have different degeneracies d_m and d_n, the N-bein is a d_m × d_n rectangular matrix, unlike Wilczek–Zee connections, which are square. Its gauge transformation accordingly uses a d_m × d_m unitary on the left and a d_n × d_n unitary on the right.

2. A two-state QGT for degenerate systems. M^{(m,n)}{ij;MN} := Σ{l≠n,m} Σ_L e^{(l,m)*}{i;LM} e^{(l,n)}{j;LN}, interpreted as encoding a transition from |n_N⟩ to |m_M⟩ through all intermediate states |l_L⟩ outside ℋ_n and ℋ_m, after two consecutive parameter variations. It is split into a symmetric part 𝒢^{(m,n)}{ij;MN} and an antisymmetric part T^{(m,n)}{ij;MN}; both are complex, gauge-dependent, matrix-valued tensors.

3. A torsion-like tensor from a Wilczek–Zee covariant derivative. The covariant derivative D_i e^{(m,n)}_{j;MN} is built with the Wilczek–Zee connections of both subspaces, and its antisymmetric part is identified as the torsion T. The paper proves that this torsion coincides exactly with the antisymmetric part of the two-state QGT — the direct analogue of the Wilczek–Zee curvature being the antisymmetric part of the degenerate QGT.

4. Gauge-invariant observables and a differential-form reformulation. The authors construct trace-type invariants (including the trace of the QGT and its real symmetric and antisymmetric parts), a q^{(n,m)}_{ij} invariant for each pair of levels, and rank-3 and rank-4 invariants built from the new tensors. They also recast the formalism in the language of differential forms and obtain a 3-form invariant reminiscent of the Chern–Simons 3-form, motivating a topological invariant analogous to the Nieh–Yan invariant.

Main Findings

• Rectangular geometry is the structural novelty of degeneracy: because d_n ≠ d_m in general, the degenerate N-bein cannot be a square matrix, and gauge transformations act independently on its left (ℋ_m) and right (ℋ_n) indices. The authors present this as a structural feature of degenerate systems rather than a technical detail.

• The N-bein is the "square root" of the QGT in the degenerate case too: the degenerate QGT is reproduced as Q^{(n)}{ij;N₁N₂} = Σ{m≠n} Σ_M e^{(m,n)*}{i;MN₁} e^{(m,n)}{j;MN₂}, and the formalism reduces to the authors' earlier non-degenerate results when d_n = 1 for all n.

• Torsion equals the antisymmetric part of the two-state QGT: explicitly, T^{(m,n)}{ij;MN} = i Σ{l≠n,m} Σ_L (e^{(m,l)}{i;ML} e^{(l,n)}{j;LN} − e^{(m,l)}{j;ML} e^{(l,n)}{i;LN}), matching the antisymmetric part of M^{(m,n)}_{ij;MN}. It captures the noncommutativity of successive parameter variations.

• Only some invariants are observables: the traces of the QGT and of its symmetric and antisymmetric parts are gauge invariant, but only the latter two are real tensors and hence counted as observables; the individual QGT trace and q^{(n,m)}_{ij} are invariant but complex.

• A Chern–Simons-like 3-form and a Nieh–Yan-like invariant: the differential-form formulation yields a 3-form invariant, and the authors state that the associated topological invariant has not previously been introduced in condensed matter and should be relevant for systems depending on three or more parameters.

• Worked example — coupled harmonic oscillators in an electric field: the coupling makes the system degenerate. The authors report all possible state transitions induced by parameter variations, covering transitions between distinct subspaces with different degeneracy as well as transitions within a single subspace, and find correlations among quantum states quantified by the new invariants.

• Some invariants carry topological information: in the example, several invariants depend only on the energy levels and are independent of the parameters.

• No numerical benchmarks, dataset sizes, or quantitative performance comparisons are reported in the supplied content; the paper is a formalism-and-example study rather than a numerical benchmark study.

Methodology in Plain English

The authors start from a Hilbert space split into orthogonal subspaces, each labeled by an energy and carrying a finite degeneracy. They track what happens to a state when the parameters it depends on are nudged slightly: part of the change stays inside the state's own degenerate subspace (described by the Wilczek–Zee connection), and part of it leaks into other subspaces. The leaking part is packaged into the N-bein.

They then borrow the logic of Cartan geometry, where the covariant derivative of the frame field gives torsion and the curvature of the connection gives the Riemann tensor. Applying this logic, the antisymmetric combination of derivatives of the N-bein gives a torsion-like object, and the curvature of the Wilczek–Zee connection gives the non-Abelian analogue of the Berry curvature.

Because these tensors change under a change of basis inside each degenerate subspace, the authors assemble combinations that are unchanged — traces over degeneracy indices — to obtain quantities that could in principle be measured. They rewrite everything with differential forms to expose topological structures, and finally test the machinery on a solvable model: two coupled harmonic oscillators in an electric field, where the coupling produces degeneracy and the invariants reveal correlations between states.

Why This Matters

Impact on research. The standard QGT is known to break down for degenerate flat bands and identical energy levels; non-Abelian tools such as the Wilczek–Zee curvature patch this gap, but a full Cartan-style frame formalism has been missing. This paper supplies that frame formalism, gives a more efficient route to computing the non-Abelian QGT, and introduces candidate topological invariants (including the Nieh–Yan analogue) that the authors argue are new to condensed matter.

Potential applications (as motivated in the paper):

• Holonomic quantum computation, where the non-Abelian quantum metric is cited as governing error-minimizing trajectories.
• Topological quantum computation, building on the fault-tolerant structures associated with non-Abelian geometric phases.
• Describing symmetry breaking and quantum phase transitions in degenerate regimes.
• Characterizing topological phases of materials, where related non-Abelian metrics have been used to derive the real Chern number of a generalized Dirac monopole and the second Chern number of a Yang monopole.

Industry relevance. The work is foundational theory, but it sits upstream of quantum-computing architectures that rely on geometric phases and of materials modeling where degeneracies (flat bands, band crossings) dominate. Its value to industry is indirect: better geometric descriptors of degenerate parameter spaces could inform how control trajectories are designed and how degenerate band structures are classified. The paper reports no device, experiment, or numerical implementation.

Future Directions

• Measuring or simulating the new invariants. The paper defines gauge-invariant observables but does not propose a concrete measurement protocol or report experimental data.
• Exploring the Nieh–Yan-like topological invariant. The authors explicitly flag this invariant as new to condensed matter and expect it to matter for quantum systems depending on three or more parameters; its physical consequences and classification power remain open.
• Applying the formalism beyond the oscillator model. The coupled-harmonic-oscillator example is the only worked case; extending it to realistic degenerate band structures, Dirac/Yang monopole settings, or flat-band systems is a natural next step.
• Connecting to non-Hermitian and PT-symmetric settings. The paper's introduction cites non-linear conductivities induced by the quantum metric in PT-symmetric systems, raising the question of whether the degenerate N-bein formalism can be adapted to non-Hermitian parameter spaces.

Target Audience

Theoretical physicists working on quantum geometry, geometric phases, and gauge structures in quantum mechanics; condensed matter theorists interested in degenerate bands, topological invariants, and non-Abelian Berry/Wilczek–Zee physics; and quantum-information researchers working on holonomic or topological quantum computation who need a geometric language for degenerate subspaces. Readers without a background in differential geometry and non-Abelian gauge theory will find the paper demanding.

Authors’ abstract

Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems; we called it $\mathcal{N}$-bein, with $\mathcal{N}$ being the number of parameters that characterize the quantum system. Acting as the ``square root'' of the quantum geometric tensor (QGT), the $\mathcal{N}$-bein allows us to define new tensors to improve our understanding of the structure beneath the parameter space of quantum mechanics. In this work, we extend this mathematical framework surrounding the $\mathcal{N}$-bein to analyze the parameter space of quantum systems with degenerate spectra. As in the non-degenerate case, we define a non-Abelian two-state QGT to identify possible transitions between degenerate states after two consecutive parameter variations. Additionally, using the Wilczek-Zee connection, we introduce a torsion-like tensor as the covariant derivative of the $\mathcal{N}$-bein. This torsion captures the noncommutativity of successive parameter variations and coincides with the antisymmetric part of the two-state QGT. We also present a geometrical formulation using differential forms and discuss the physical implications of the newly defined tensors. Furthermore, we construct several gauge-invariant observables from the $\mathcal{N}$-bein and its derivatives to highlight the utility of the new tensors. Finally, to illustrate the convenience and applications of this formalism, we apply the theoretical framework to a system of coupled harmonic oscillators immersed in an electric field. The coupling between the oscillators results in a degenerate system. Thus, using the new formalism, we found correlations among the quantum states quantified by the new invariants.