# Definition Consider a system that depends on some time-dependent [[vector]] $\pmb{x}(t)$. We assume the [[Hamiltonian]] is $H[\pmb{x}(t)]$, and the Hamiltonian's $n^{\text{th}}$ [[Eigenstate]] is $\ket{n, \pmb{x}(t)}$. The [[Schrödinger Equation]] for this system is: $ \begin{align} H[\pmb{x}(t)] \ket{n, \pmb{x}(t)} = E_n[\pmb{x}(t)] \ket{n, \pmb{x}(t)} .\end{align} $ We assume $\pmb{x}(t=0) = \pmb{x}_0$ and that $\pmb{x}(t)$ changes [[Adiabatic|adiabatically]] in time. From the time-dependent [[Schrödinger Equation]]: $ \begin{align} H[\pmb{x}(t)] \ket{n, t} = i\hbar \frac{\partial}{\partial t} \ket{n, t}, \end{align} $ we can obtain the [[Eigenstate]] at time $t$: $ \begin{align} \ket{n,t} = &\exp\left(\frac{i}{\hbar}\int_0^t \text{d} t' E_n[\pmb{x}(t')]\right) \exp(-\gamma[\pmb{x}(t)]) \ket{n, \pmb{R}(t)} , \end{align} $ where $ \begin{align} \gamma[\pmb{x}(t)] \equiv \int_0^t \text{d} t'\left(i \pmb{\dot{x}}(t') \cdot \bra{n, \pmb{x}(t')} \nabla_{\pmb{x}} \ket{n, \pmb{x}(t')}\right). \end{align} $ The phase of the first term in the equation for $\ket{n, t}$ is the trivial dynamical phase, and $\gamma$ is known as the nontrivial geometric phase, Berry phase or Panchanatram phase. Now assume that the [[vector]] $\pmb{x}(t)$ traces a closed [[Contour]] $C$ from $t = 0$ to $t=T$, i.e. $\pmb{x}(t=0) = \pmb{x}(t=T) = \pmb{x}_0$. We label the area enclosed by the contour $S$. Then we can simplify $\gamma$ as follows: $ \begin{align} \gamma[C] &\equiv \int_0^T \text{d} t\left(i \pmb{\dot{x}}(t) \cdot \bra{n, \pmb{x}(t)} \nabla_{\pmb{x}} \ket{n, \pmb{x}(t)}\right), \\ &= \oint_C \text{d} \pmb{x} \cdot i \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{F}_n(\pmb{x}), \end{align} $ where in the last equation we used [[Stokes' theorem]]. Here we have defined the [[Berry Connection]] $\pmb{A}_n(\pmb{x})$ and the [[Berry Curvature]] $\pmb{\Omega}_n(\pmb{x})$ of the $n^{\text{th}}$ [[Band Structure|band]]: $ \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} $ # Gauge Invariance Note that the Berry phase is [[gauge]] invariant modulo $2\pi$: $ \begin{align} \ket{n, \pmb{x}} &\rightarrow e^{-i \beta(\pmb{x})} \ket{n, \pmb{x}}, \\ \gamma[C] &\rightarrow \gamma[C] + \beta(t=T) - \beta(t=0) \\ \end{align} $ By continuity of the close [[Contour]] $C$, we have $\beta(T) - \beta(0) = 2 \pi n$ where $n \in \mathbb{Z}$. # Applications The Berry phase has many applications in physics, including the study of [[Topological systems]].