# Definition Consider the [[Berry phase]] of a system that depends on a [[vector]] $\pmb{x}(t)$ that varies [[adiabatic|adiabatically]] in time. Assume the vector evolves cyclically along a closed [[Contour]] $C$, then the [[Berry phase]] is: $ \begin{align} \gamma[C] &= i \oint_C \text{d} \pmb{x} \cdot \bra{n, \pmb{x}(t)} \nabla_{\pmb{x}} \ket{n, \pmb{x}(t)},\\ &\equiv \oint_C \text{d} \pmb{x} \cdot \pmb{A}_n(\pmb{x}),\\ &\equiv \int_S \text{d} \pmb{S} \cdot \pmb{\Omega}_n(\pmb{x}), \end{align} $ Where we have defined the [[Berry Connection]] and Berry curvature respectively as follows: $ \begin{align} \pmb{A}_n(\pmb{x}) &\equiv i \bra{n,\pmb{x}} \nabla_{\pmb{x}} \ket{n,\pmb{x}},\\ \pmb{\Omega}_n(\pmb{x}) &\equiv \nabla_{\pmb{x}} \times \pmb{A}_n(\pmb{x}) .\end{align} $ More generally, it is the [[skew-symmetric matrix|anti-symmetric]] second-rank [[tensor]] computed from the Berry connection as $ \begin{align} \Omega_{n, \mu\nu}(\pmb{x}) &= \frac{\partial}{\partial x^\mu} A_{n, \nu} (\pmb{x}) - \frac{\partial}{\partial x^\nu} A_{n, \mu} (\pmb{x}), \\ \Omega_{n, \mu\nu}(\pmb{x}) &= \epsilon_{\mu\nu\xi} \pmb{\Omega}_{n, \xi}(\pmb{x}) \end{align} $ # Gauge Invariance Note that the Berry curvature itself is [[gauge]] invariant, unlike the [[Berry Connection]] which must be integrated along a closed path to become physical. This can be seen easily because $ \begin{align} \ket{n, \pmb{x}} &\rightarrow e^{-i \beta(\pmb{x})} \ket{n, \pmb{x}}, \\ \gamma[C] &\rightarrow \gamma[C] + 2 \pi n, \quad n \in \mathbb{Z}, \\ \pmb{A}_n(\pmb{x}) &\rightarrow \pmb{A}_n(\pmb{x}) + \nabla_{\pmb{x}} \beta(\pmb{x}),\\ \pmb{\Omega}_n(\pmb{x}) &\rightarrow \pmb{\Omega}_n(\pmb{x}) + \nabla_{\pmb{x}} \times \nabla_{\pmb{x}} \beta(\pmb{x}) = \pmb{\Omega}_n(\pmb{x}) \end{align} $ # Properties Under Symmetry In systems with [[Time-Reversal Operator|time-reversal symmetry]], it is well known that $\pmb{\Omega}(-\pmb{k}) = -\pmb{\Omega}(\pmb{k})$. In systems with [[Inversion Operator|inversion symmetry]], we additionally have $\pmb{\Omega}(-\pmb{k}) = +\pmb{\Omega}(\pmb{k})$. Thus, in systems with both TRS and IS, the Berry Curvature is identically zero $\pmb{\Omega}(\pmb{k}) \equiv 0$. # Physical Picture The Berry curvature is a local manifestation of the geometric properties of the [[Wavefunction|wavefunctions]] in the parameter space.