Accepted Paper
Inserted: 23 dec 2025
Last Updated: 25 jun 2026
Journal: The Journal of Geometric Analysis (Springer)
Year: 2026
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 $M=T^{2}_{\mathrm{BZ}}\times S^{1}_{\phi_{+}}\times S^{1}_{\phi_{-}}$ carries a Kaluza--Klein metric $g_{M}$ and a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part defines a mixed cohomology class $ [\omega]\in\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{+}})\bigr)\oplus\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{-}})\bigr), $ whose mixed tensor rank equals one. By adapting the \textit{Pigazzini--Toda lower bound} to the Kaluza--Klein setting through exact pointwise curvature analysis under the assumption of constant Berry curvatures, we show that the obstruction kernel $\mathcal{K}$ vanishes and establish a three-level non-reducibility structure for the physical metric: $\textit{(i)}$ for the one-parameter deformation family interpolating between the product and physical metrics, $\text{dim}\mathfrak{hol}^{\mathrm{off}}(\nabla^{C_\varepsilon})\geq 1$ at every point for all $\varepsilon\in(0,1)$; $\textit{(ii)}$ at the physical metric, every non-Bismut torsion representative of $[\omega]$ yields $\text{dim}\mathfrak{hol}^{\mathrm{off}}\geq 1$ on an open set; $\textit{(iii)}$ the horizontal--vertical splitting is not invariant under the Riemannian holonomy of the physical metric, with $\text{dim}\mathfrak{hol}^{\mathrm{off}}(\nabla^{\mathrm{LC}})\geq 1$ at every point. These bounds prevent the complete gauging-away of Berry phases even in regimes with zero net topological charge. The corrected rank $r^{\sharp}$ detects the robustness of the topological constraint under phase-reduction protocols: no single phase-locking can eliminate the obstruction at the physical metric, a distinction invisible to the mixed rank $r$ alone. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.
Keywords: Berry curvature, spin--orbit-coupled Bose--Einstein condensates, synthetic gauge fields, holonomy with torsion, mixed cohomology, Kaluza--Klein geometry, PT lower bound
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