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 $M=T^{2}_{\mathrm{BZ}}\times S^{1}{φ{+}}\times S^{1}{φ{-}}$ carries a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part 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.