The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as H = μ μ σ σ ⋅ ⋅ B , where σ σ denote the Pauli matrices, μ μ is the magnetic moment, and B is the magnetic field. In three dimensions, the eigenstates have energies ± ± μ μ B and their eigenvectors are | u − − ⟩ ⟩ = ( sin ⁡ ⁡ θ θ 2 e − − i ϕ ϕ − − cos ⁡ ⁡ θ θ 2 ) , | u + ⟩ ⟩ = ( cos ⁡ ⁡ θ θ 2 e − − i ϕ ϕ sin ⁡ ⁡ θ θ 2 ) . Now consider the | u − − ⟩ ⟩ state. Its Berry connection can be computed as A θ θ = ⟨ ⟨ u − − | i 1 r ∂ ∂ θ θ | u − − ⟩ ⟩ = 0 , A ϕ ϕ = ⟨ ⟨ u − − | i 1 r sin ⁡ ⁡ θ θ ∂ ∂ ϕ ϕ | u − − ⟩ ⟩ = sin 2 ⁡ ⁡ θ θ 2 r sin ⁡ ⁡ θ θ , and the Berry curvature is Ω Ω r = 1 r sin ⁡ ⁡ θ θ [ ∂ ∂ θ θ ( A ϕ ϕ sin ⁡ ⁡ θ θ ) − − ∂ ∂ ϕ ϕ A θ θ ] r ^ ^ = 1 2 r 2 r ^ ^ . If we choose a new gauge by multiplying | u − − ⟩ ⟩ by e i ϕ ϕ (or any other phase e i α α ϕ ϕ , α α ∈ ∈ R ), the Berry connections are A θ θ = 0 and A ϕ ϕ = − − cos 2 ⁡ ⁡ θ θ 2 r sin ⁡ ⁡ θ θ , while the Berry curvature remains the same. This is consistent with the conclusion that the Berry connection is gauge-dependent while the Berry curvature is not.