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 $M=T_{BZ}^2\times S_{{\varphi }_{+}}^1\times S_{{\varphi }_{-}}^1$ 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 in $(H^2(T_{BZ}^2)\otimes H^1(S_{{\varphi }_{+}}^1))\oplus (H^2(T_{BZ}^2)\otimes H^1(S_{{\varphi }_{-}}^1))$ of mixed tensor rank one. Adapting the Pigazzini-Toda lower bound to the Kaluza-Klein setting through exact pointwise curvature analysis (constant Berry curvatures), we show that the obstruction kernel vanishes and obtain a three-level non-reducibility structure for the physical metric: (i) for the one-parameter family interpolating between the product and physical metrics, $\dim {\mathfrak{hol}}^{\mathrm{off}}({\nabla }^{C_{\epsilon }})\ge 1$ at every point for all $\epsilon \in (0,1)$; (ii) at the physical metric, every non-Bismut torsion representative of $[\omega ]$ yields $\dim {\mathfrak{hol}}^{\mathrm{off}}\ge 1$ on an open set; (iii) the horizontal-vertical splitting is not invariant under the Riemannian holonomy of the physical metric, with $\dim {\mathfrak{hol}}^{\mathrm{off}}({\nabla }^{\mathrm{LC}})\ge 1$ at every point. These bounds prevent the complete gauging-away of Berry phases even at zero net topological charge. The corrected rank $r^{♯}$ detects the robustness of the constraint under phase-reduction protocols: no single phase-locking can eliminate the obstruction, 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.

v6: corrected the off-diagonal Levi-Civita curvature on mixed inputs (previously stated as zero), giving the exact factor $({\epsilon }^2-1)$; at the physical metric the natural (Bismut) torsion has vanishing off-diagonal curvature, and the bound is re-established via non-Bismut representatives and the Riemannian holonomy. Holonomy=non-invariance of the $H\oplus V$ splitting.Main results unchanged

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• Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates
• Observation of Dirac Monopoles in a Synthetic Magnetic Field
• An Aharonov-Bohm interferometer for determining Bloch band topology
• A large magnetic storage ring for Bose-Einstein condensates