Amplitude-Weighted Topological Protection in Cold-Atom Bose-Hubbard Lattices: A Critical Interaction Strength for Direction-Field Vortices

Using exact diagonalization on a 6x6 soft-core Bose-Hubbard model at fractional magnetic flux Φ = Φ_0/q, we identify a critical interaction strength U* that destroys the topological protection of a quantum direction-field vortex. We investigate this breakdown through a direction-field elastic energy functional E[u] = Σ w_ij (1 - u_i · u_j) with amplitude weights w_ij = |φ_i||φ_j|. For the integer-flux case q=1, whose ground-state direction field carries a central vortex with winding number C_h = +1, we find a critical interaction strength U* between 8 and 9: for U < U, Riemannian gradient descent converges to a vortex-bearing local minimum (E ≈ 0.44, C_h = +1); for U > U, it slides into a vortex-free global minimum (E = 0, C_h = 0). The transition is not a vortex–antivortex annihilation but a single-texture unwinding: one plaquette edge's phase difference crosses the ±π boundary, and the winding number jumps from +1 to 0 at an energy cost of ΔE ≈ 0.008. An amplitude-free control functional preserves the vortex for all U and yields identical Hessian spectra across U=1, 5, 10, demonstrating that topological protection in this setting originates from the functional structure lacking a vortex-free minimum, not from amplitude weighting. The Hessian at the vortex-bearing critical point exhibits three zero modes (vortex translation) and one strong negative mode (vortex-core direction reversal), confirming Morse–Bott degeneracy with saddle-point character. For q ≥ 2, the ground-state direction field is smooth (C_h = 0), the gradient flow preserves zero winding from the first iteration, and the concept of U* does not apply. These results identify amplitude anchoring as a conditional truncation of topological protection, with direct implications for the design of direction-field probes in cold-atom experiments. We also discuss the framework extension of cold-atom systems as a physical instance of the O1 Diagnoser (states 1 and ⊥) and its ontological distinction from topological quantum computing.