Berry位相
Journal of Physics A: Mathematical and General 18 (2): 221–230. doi:10.1088/0305-4470/18/2/011. ISSN 0305-4470. https://iopscience.iop.org/article/10.1088/0305-4470/18/2/011.
↑ N. A. Sinitsyn; I. Nemenman (2007). “The Berry phase and the pump flux in stochastic chemical kinetics”. Europhysics Letters 77 (5): 58001. arXiv:q-bio/0612018. Bibcode: 2007EL.....7758001S. doi:10.1209/0295-5075/77/58001.
↑ Xiao, Di; Chang, Ming-Che; Niu, Qian (2010-07-06). “Berry phase effects on electronic properties”. Reviews of Modern Physics 82 (3): 1959–2007. doi:10.1103/RevModPhys.82.1959. https://link.aps.org/doi/10.1103/RevModPhys.82.1959.
Gaugeの独立した、Berry接続は
A n ( R ) = i ⟨ n ( R ) | ∇ R | n ( R ) ⟩ {\displaystyle {\mathcal {A}}_{n}(\mathbf {R} )=i\langle n(\mathbf {R} )|\nabla _{\mathbf {R} }|n(\mathbf {R} )\rangle }
だから
γ n = ∫ C d R ⋅ A n ( R ) {\displaystyle \gamma _{n}=\int _{\mathcal {C}}d\mathbf {R} \cdot {\mathcal {A}}_{n}(\mathbf {R} )}
ストークスの定理のために
γ n = ∫ S d S ⋅ Ω n ( R ) . {\displaystyle \gamma _{n}=\int _{\mathcal {S}}d\mathbf {S} \cdot \mathbf {\Omega } _{n}(\mathbf {R} ).}
そこ、Berry曲率は
Ω n , μ ν ( R ) = ∂ ∂ R μ A n , ν ( R ) − ∂ ∂ R ν A n , μ ( R ) . {\displaystyle \Omega _{n,\mu \nu }(\mathbf {R} )={\partial \over \partial R^{\mu }}{\mathcal {A}}_{n,\nu }(\mathbf {R} )-{\partial \over \partial R^{\nu }}{\mathcal {A}}_{n,\mu }(\mathbf {R} ).}