Accepted Paper
Inserted: 23 dec 2025
Last Updated: 20 apr 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 irreducibility structure for the physical metric: (i) for the one-parameter deformation family interpolating between the product and physical metrics, \(\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C_\varepsilon})\geq 1\) at every point for all \(\varepsilon\in(0,1)\); (ii) at the physical metric, every non-Bismut torsion representative of \([\omega]\) yields \(\dim\mathfrak{hol}^{\mathrm{off}}\geq 1\) on an open set; (iii) the Riemannian holonomy of \(g_M\) satisfies \(\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{\mathrm{LC}})\geq 1\) 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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