preprint
Inserted: 23 dec 2025
Last Updated: 18 mar 2026
Year: 2025
Abstract:
We establish a local topological obstruction to the simultaneous flattening of Berry curvature in spin--orbit-coupled Bose--Einstein condensates (SOC BECs), which remains valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space is the total space \(M\) of a principal \(U(1)_+ \times U(1)_-\) bundle over the Brillouin torus \(T^{2}_{\mathrm{BZ}}\). On \(M\), we construct a Kaluza--Klein metric and a natural metric connection \(\nabla^{C}\) whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class \[ [\omega] \in \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{+}})\bigr) \oplus \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{-}})\bigr) \] with mixed tensor rank \(r=1\). By adapting the \textit{Pigazzini--Toda (PT) lower bound} to the Kaluza--Klein setting through exact pointwise curvature analysis, we demonstrate that the obstruction kernel \(\mathcal{K}\) vanishes for the physical metric, yielding the cohomological invariant \(r^{\sharp}=1\). The exact curvature formula reveals a three-level irreducibility structure: (i) for the one-parameter deformation family \(g_\varepsilon\) interpolating between the product metric and the physical Kaluza--Klein metric, \(\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C_\varepsilon})\geq 1\) holds at every point for all \(\varepsilon\in(0,1)\); (ii) at the physical endpoint \(\varepsilon=1\), the natural Kaluza--Klein torsion---identified as the Bismut torsion---produces an exact cancellation of the off-diagonal curvature, which we characterise as a non-generic phenomenon: every other torsion representative of \([\omega]\) yields \(\dim\mathfrak{hol}^{\mathrm{off}}\geq 1\) on an open non-empty set; (iii) the Riemannian holonomy of the physical metric is itself irreducible, with \(\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{\mathrm{LC}})\geq 1\) at every point. These bounds prevent the complete gauging-away of Berry phases even in regimes with zero net topological charge. This work provides the first cohomological lower bound, based on the PT framework, certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm, and identifies the Bismut cancellation as the unique obstruction to extending the bound from the deformation family to the natural Kaluza--Klein connection.
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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