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

• URL: http://arxiv.org/abs/2609.01752v1
• Date: Tue, 01 Sep 2026 18:22:09 GMT
• Title: $\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry
• Abstract summary: Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems.<n>We extend this framework to analyze the parameter space of quantum systems with degenerate spectra.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Typical entanglement in anyon chains: Page curves beyond Lie group symmetries [41.99844472131922]
  We study bipartite entanglement statistics in one-dimensional anyon chains constrained by fusion rules of unitary pre-modular categories.<n>Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(sqrtL)$ or $O(1)$ symmetry-type corrections.<n> Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions.
  arXiv  Detail & Related papers  (2026-03-26T18:00:03Z)
• Hysteretic squashed entanglement in many-body quantum systems [42.085941481155295]
  Entanglement in many-body quantum systems is distributed across spatial regions.<n>We propose hysteretic entanglement $T_sq$, a conditional entanglement that measures the genuine quantum correlations between two subregions.<n>We show that $T_sq$ can detect genuine quantum correlations across both adjacent and long-range subsystems.
  arXiv  Detail & Related papers  (2026-03-10T17:00:49Z)
• Continuum field theory of matchgate tensor network ensembles [36.94429692322632]
  We develop a continuum description for random ensembles of two-dimensional fermionic matchgate tensor networks.<n>As a diagnostic of the resulting universal physics, we analyze disorder-averaged moments of fermionic two-point functions.<n>We show that disorder drives universal long-distance behavior governed by a nonlinear sigma-model of symmetry class D with a topological term.
  arXiv  Detail & Related papers  (2026-03-06T12:15:35Z)
• Meson spectroscopy of exotic symmetries of Ising criticality in Rydberg atom arrays [39.58317527488534]
  Coupling two Ising chains in a ladder leads to an even richer $mathcalD(1)_8$ symmetries.<n>Here, we probe these emergent symmetries in a Rydberg atom processing unit, leveraging its geometry to realize both chain and ladder configurations.
  arXiv  Detail & Related papers  (2025-06-26T14:19:30Z)
• Doubly Quantum Mechanics [0.0]
  We develop the formalism for spin-$frac12$ measurements by promoting the group of spatial rotations $SU(2)$ to the quantum group $SU_q(2)$.<n>We find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of $SU_q(2)$.
  arXiv  Detail & Related papers  (2024-12-08T17:10:35Z)
• Hilbert space geometry and quantum chaos [39.58317527488534]
  We consider the symmetric part of the QGT for various multi-parametric random matrix Hamiltonians. We find for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect.
  arXiv  Detail & Related papers  (2024-11-18T19:00:17Z)
• $N$-bein formalism for the parameter space of quantum geometry [0.0]
  This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it $N$-bein. 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. We find that the new tensors quantify correlations between quantum states that were unavailable by other methods.
  arXiv  Detail & Related papers  (2024-06-27T18:16:11Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find 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.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Quantum and classical spin network algorithms for $q$-deformed Kogut-Susskind gauge theories [0.0]
  We introduce $q$-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. Our proposal simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. Our work gives a new perspective for the application of tensor network methods to high-energy physics.
  arXiv  Detail & Related papers  (2023-04-05T15:49:20Z)
• Shape Invariant Potentials in Supersymmetric Quantum Cosmology [0.0]
  shape invariant potentials is an essential feature in many settings of $N=2$ supersymmetric quantum mechanics. We show how shape in property can be employed to bring a relation among several factor orderings choices for our Schr"odinger-Wheeler-DeWitt equation.
  arXiv  Detail & Related papers  (2022-05-31T19:51:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.