Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.01752 · Journal of Physics A Mathematical and Theoretical

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

Jorge Romero, Carlos A Velasquez, J David Vergara

quant-phcond-mat.othergr-qcSubmitted Sep 1, 2026• Announced Sep 3, 2026• 0 citations
arXivHTMLPDF

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.

Journal reference
Jorge Romero et al 2026 J. Phys. A: Math. Theor. 59 355305
DOI
10.1088/1751-8121/ae9c7a
Comments
33 pages, 7 figures. Accepted version (Journal of Physics A: Mathematical and Theoretical)

Related papers

N-bein formalism for the parameter space of quantum geometry

Jorge Romero, Carlos A. Velasquez, J. David Vergara

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 à 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.

arXiv:2406.19468•Journal of Physics A Mathematical and Theoretical•Aug 14, 2024•5 citations

Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

Diego González, Daniel Gutiérrez‐Ruiz, J. David Vergara

Abstract The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek–Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.

arXiv:2011.14310•Journal of Physics A Mathematical and Theoretical•Nov 2, 2020•15 citations

Time-dependent quantum geometric tensor and some applications

Bogar Díaz, Diego González, Marcos J. Hernández +1 more

Abstract We define a time-dependent extension of the quantum geometric tensor to describe the geometry of the time-parameter space for a quantum state by considering small variations in both time and wave function parameters. Compared to the standard quantum geometric tensor, this tensor introduces new temporal components, enabling the analysis of systems with non-time-separable or explicitly time-dependent quantum states and encoding new information about these systems. In particular, the time-time component of this tensor is related to the energy dispersion of the system. We applied this framework to a harmonic/inverted oscillator, a time-dependent harmonic oscillator, and a chain of generalized harmonic/inverted oscillators. We show some results on the scalar curvature associated with the time-dependent quantum geometric tensor and the generalized Berry curvature behavior on the transition from harmonic oscillators to inverted ones. Furthermore, we analyze the entanglement for the chain through purity analysis, obtaining that the purity for any excited state is zero in the mentioned transitions.

arXiv:2502.01788•Physica Scripta•Jun 26, 2025•3 citations

Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians

Alexander Avdoshkin

Understanding the geometric information contained in quantum states is valuable in various branches of physics, particularly in solid-state physics when Bloch states play a crucial role. While the Fubini-Study metric and Berry curvature form offer comprehensive descriptions of non-degenerate quantum states, a similar description for degenerate states did not exist. In this work, we fill this gap by showing how to reduce the geometry of degenerate states to the non-abelian (Wilczek-Zee) connection $A$ and a previously unexplored matrix-valued metric tensor $G$. Mathematically, this problem is equivalent to finding the $U(N)$ invariants of a configuration of subspaces in $\mathbb{C}^n$. For two subspaces, the configuration was known to be described by a set of $m$ principal angles that generalize the notion of quantum distance. For more subspaces, we find $3 m^2 - 3 m + 1$ additional independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. We also develop a procedure for calculating these invariants as integrals of $A$ and $G$ over geodesics on the Grassmannain manifold. Finally, we briefly discuss possible application of these results to quantum state preparation and $PT$-symmetric band structures.

arXiv:2404.03234•Apr 4, 2024•1 citation

Operator-frame geometry of non-compact quantum systems with frame-vacuum phase transitions

Satoshi Tanaka, Gonzalo Ordonez, Masatoshi Kubota +2 more

We formulate the geometric structure of non-compact bosonic quantum systems in regimes where vacuum instability renders the relevant quantum states non-normalizable, causing conventional state-space quantum geometry -- described by the Berry connection, curvature, and quantum metric -- to become ill-defined. To overcome this breakdown, we develop a formulation of quantum geometry at the level of canonical operator frames, allowing for complexified Bogoliubov-Valatin transformations that lift the requirement that creation operators be Hermitian conjugates of annihilation operators. Canonical operator frames are defined as choices of bosonic creation and annihilation operators realizing the canonical commutation relations. A natural equivalence relation among such frames generalizes the phase ambiguity of quantum states and determines a parameter space that analytically extends the stable-regime parameter space. The space of canonical operator frames forms a principal bundle over parameter space -- the operator-frame bundle -- equipped with a natural Ehresmann connection that defines parallel transport while preserving the canonical commutation relations. In the stable regime, this construction reduces to the Berry connection, while more generally it yields a well-defined operator-space quantum geometric tensor (QGT) that remains valid across vacuum instabilities. Using the framework of rigged Hilbert spaces, we define a notion of quantum frame vacuum and obtain a consistent state-space QGT. Focusing on a single bosonic mode, we demonstrate analyticity of the QGT across the quantum frame-vacuum phase transition and present the corresponding phase diagram on the complexified squeezing-parameter plane. We further introduce a realistic physical setting that allows continuous paths connecting stable and unstable regimes, along which the QGT evolves smoothly across Stokes lines.

quant-pharXiv:2607.06994•Jul 9, 2026•0 citations

Quantum geometric tensors from sub-bundle geometry

Marius A. Oancea, Thomas Mieling, Giandomenico Palumbo

The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.

arXiv:2503.17163•Quantum•Jan 14, 2026•2 citations
N\mathcal{N}N-bein formalism for degenerate states in the parameter space of quantum geometry
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 N\mathcal{N}N-bein, with N\mathcal{N}N being the number of parameters that characterize the quantum system. Acting as the ``square root'' of the quantum geometric tensor (QGT), the N\mathcal{N}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 N\mathcal{N}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 N\mathcal{N}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 N\mathcal{N}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.
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 à 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.
Abstract The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek–Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.
Abstract We define a time-dependent extension of the quantum geometric tensor to describe the geometry of the time-parameter space for a quantum state by considering small variations in both time and wave function parameters. Compared to the standard quantum geometric tensor, this tensor introduces new temporal components, enabling the analysis of systems with non-time-separable or explicitly time-dependent quantum states and encoding new information about these systems. In particular, the time-time component of this tensor is related to the energy dispersion of the system. We applied this framework to a harmonic/inverted oscillator, a time-dependent harmonic oscillator, and a chain of generalized harmonic/inverted oscillators. We show some results on the scalar curvature associated with the time-dependent quantum geometric tensor and the generalized Berry curvature behavior on the transition from harmonic oscillators to inverted ones. Furthermore, we analyze the entanglement for the chain through purity analysis, obtaining that the purity for any excited state is zero in the mentioned transitions.
Geometry of degenerate quantum states, configurations of mmm-planes and invariants on complex Grassmannians
Understanding the geometric information contained in quantum states is valuable in various branches of physics, particularly in solid-state physics when Bloch states play a crucial role. While the Fubini-Study metric and Berry curvature form offer comprehensive descriptions of non-degenerate quantum states, a similar description for degenerate states did not exist. In this work, we fill this gap by showing how to reduce the geometry of degenerate states to the non-abelian (Wilczek-Zee) connection AAA and a previously unexplored matrix-valued metric tensor GGG. Mathematically, this problem is equivalent to finding the U(N)U(N)U(N) invariants of a configuration of subspaces in Cn\mathbb{C}^nCn. For two subspaces, the configuration was known to be described by a set of mmm principal angles that generalize the notion of quantum distance. For more subspaces, we find 3m2−3m+13 m^2 - 3 m + 13m2−3m+1 additional independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. We also develop a procedure for calculating these invariants as integrals of AAA and GGG over geodesics on the Grassmannain manifold. Finally, we briefly discuss possible application of these results to quantum state preparation and PTPTPT-symmetric band structures.
We formulate the geometric structure of non-compact bosonic quantum systems in regimes where vacuum instability renders the relevant quantum states non-normalizable, causing conventional state-space quantum geometry -- described by the Berry connection, curvature, and quantum metric -- to become ill-defined. To overcome this breakdown, we develop a formulation of quantum geometry at the level of canonical operator frames, allowing for complexified Bogoliubov-Valatin transformations that lift the requirement that creation operators be Hermitian conjugates of annihilation operators. Canonical operator frames are defined as choices of bosonic creation and annihilation operators realizing the canonical commutation relations. A natural equivalence relation among such frames generalizes the phase ambiguity of quantum states and determines a parameter space that analytically extends the stable-regime parameter space. The space of canonical operator frames forms a principal bundle over parameter space -- the operator-frame bundle -- equipped with a natural Ehresmann connection that defines parallel transport while preserving the canonical commutation relations. In the stable regime, this construction reduces to the Berry connection, while more generally it yields a well-defined operator-space quantum geometric tensor (QGT) that remains valid across vacuum instabilities. Using the framework of rigged Hilbert spaces, we define a notion of quantum frame vacuum and obtain a consistent state-space QGT. Focusing on a single bosonic mode, we demonstrate analyticity of the QGT across the quantum frame-vacuum phase transition and present the corresponding phase diagram on the complexified squeezing-parameter plane. We further introduce a realistic physical setting that allows continuous paths connecting stable and unstable regimes, along which the QGT evolves smoothly across Stokes lines.
The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.