# Berry curvature and linear response functions

Let $\hat{A}^i (i = 1, . . . , n)$ be a set of hermitian observables and $F_i$ a corresponding set of external fields that are linearly coupled to $\hat{A}^i$. Starting from the ground-state at $F_i = 0$ imagine a process in which the $F_i$ ’s are slowly changed (always remaining small) around the closed loop $$0\rightarrow \delta F_i^{(1)}\rightarrow \delta F_i^{(2)}\rightarrow \delta F_i^{(1)}\rightarrow 0$$ where $\delta F_i^{(1)}$ and $\delta F_i^{(2)}$ are two sets of small increments. At the end of the cycle the system will return to the ground-state having acquired a 'Berry phase' $e^{i\gamma}$ where $$\gamma=\sum_{ij}\delta F_{i}^{(1)}\Omega_{ij}\delta F_{j}^{(2)}$$ and $$\Omega_{ij}\equiv 2\Im m \langle \frac{\partial \psi}{\partial F_i} | \frac{\partial \psi}{\partial F_j} \rangle$$ is the Berry curvature. Here $$| \frac{\partial \psi}{\partial F_j} \rangle =\frac{\partial }{\partial F_j} |\psi\rangle,$$ where $| \psi \rangle$ is the state of the system regarded as a functional of the $F_i$. By evaluating $\frac{\partial }{\partial F_0} |\psi\rangle$ by first-order perturbation theory and comparing the resulting expression for $\Omega_{ij}$ with the Lehmann representation

for the linear response function $\chi_{A_i A_j}(\omega)$ (at $T = 0$) show that the Berry curvature can be expressed in the following manner:

• Ref: Gabriele Giuliani,"QUANTUM THEORY OF THE ELECTRON LIQUID" – Abolfazl Dec 10 '14 at 17:45
• the point and my problem mostly is $| \frac{\partial \psi}{\partial F_j} \rangle$ – Abolfazl Dec 11 '14 at 19:00