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 $M=T^2_{BZ}\times S^1_{ϕ_+}\times S^1_{ϕ_-}$ 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 $(H^2(T^2_{BZ})\otimes H^1(S^1_{ϕ_+}))\oplus(H^2(T^2_{BZ})\otimes H^1(S^1_{ϕ_-}))$ 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, $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C_\varepsilon})\ge1$ at every point for all $\varepsilon\in(0,1)$; (ii) at the physical metric, every non-Bismut torsion representative of $[ω]$ yields $\dim\mathfrak{hol}^{\mathrm{off}}\ge1$ on an open set; (iii) the horizontal-vertical splitting is not invariant under the Riemannian holonomy of the physical metric, with $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{\mathrm{LC}})\ge1$ at every point. These bounds prevent the complete gauging-away of Berry phases even at zero net topological charge. The corrected rank $r^\sharp$ 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.