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Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs

Alexander Pigazzini, Magdalena Toda

Abstract

We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space \(M=T^{2}_{\mathrm{BZ}}\times S^{1}_{φ_{+}}\times S^{1}_{φ_{-}}\) carries a natural metric connection \(\nabla^{C}\) whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class $$ [ω]\in\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{φ_{+}})\bigr)\oplus\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{φ_{-}})\bigr), $$ whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel $\mathcal{K}$ vanishes, yielding the sharp inequality $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C})\geq 1$. This forces at least one off-diagonal curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.

Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs

Abstract

We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space carries a natural metric connection whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel vanishes, yielding the sharp inequality . This forces at least one off-diagonal curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.
Paper Structure (36 sections, 12 theorems, 102 equations)

This paper contains 36 sections, 12 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Geometric Construction of the Extended Parameter Space
  3. Global structure of the extended parameter space.
  4. Kaluza--Klein metric on the extended space
  5. Metric connection with prescribed torsion
  6. Parallel forms in the Kaluza--Klein geometry
  7. Summary of the geometric setup
  8. Cohomology of the Extended Space and the Mixed Tensor Rank
  9. Künneth decomposition of $H^{3}(M)$
  10. Mixed tensor rank of the torsion class
  11. Vanishing of the obstruction kernel $\mathcal{K}$
  12. Summary of the topological data
  13. Geometric foundations for the PT theorem in Kaluza--Klein geometry
  14. Deformation to a product metric and stability
  15. Persistence of non‑parallelism under metric change
  16. ...and 21 more sections

Key Result

Lemma 2.2

The metric eq:KKmetric satisfies the following.

Theorems & Definitions (34)

  • Lemma 2.2: Properties of the metric $g_{M}$
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5: Triviality of the obstruction kernel $\mathcal{K}$ under constant curvature
  • proof
  • Remark 2.6: Physical relevance of constant Berry curvatures
  • Lemma 3.1: Explicit harmonic representative
  • proof
  • Remark 3.2
  • ...and 24 more