Berry Phase for Bloch electrons I am new to the topic of Berry phase.
The definition says that Berry phase depends only on the path in the parameter space of $R$, where the Hamiltonian is $H(R)$, but whatever problems I have seen, the parameter itself has a time dependence. Even for the case of Bloch electrons, we can calculate Berry phase for cyclic excursion in the parameter space $k$ of the lattice. 
The real space in the lattice is absolutely time independent; my question is that will there be a Berry phase, if we perform a cyclic excursion of an electron in the real space of the lattice? 
 A: In real space, the adiabatic Berry phase of a closed orbit just measures the magnetic flux through the orbit's area.
Explanation: (please see Sundaram and Niu )
The semiclassical equations of motion of a Block electron in phase space are given by (Sundaram and Niu equation 3.8)
$$\mathbf{\dot{x}_c} = \frac{\partial \mathcal {E}_M}{\partial \mathbf{k_c} }-\mathbf{\dot{k}_c} \times\mathbf{ \Omega}$$
$$\mathbf{\dot{k}_c} = -e \mathbf{E} - \mathbf{\dot{x}_c} \times\mathbf{ B}$$
Where:  $\mathbf{x_c}$, $\mathbf{k_c}$, are the electron wavepacket center of mass position and momentum respectively,  $\mathcal{E}_M$ is the magnetic Bloch band energy,  $\mathbf{ E}$ and $\mathbf{ B}$ are the electric and magnetic fields respectively and $\mathbf{ \Omega}$ is the berry curvature.
These equations of motion can be obtained from the Lagrangian (Sundaram and Niu equation 3.7):
$$L = \mathcal {E}_M(\mathbf{k_c}) + e \phi(\mathbf{x_c})+  \mathbf{\dot{x}_c}\cdot\mathbf{k_c} – e  \mathbf{\dot{x}_c} \cdot \mathbf{A}  +  \mathbf{\dot{k}_c} \cdot \mathbf{A}_B  $$
Where $\phi$ is the electromagnetic  scalar potential  $\mathbf{A} $ is the electromagnetic vector potential  and $\mathbf{A} _B$  is the Berry potential.
Please observe that the above formulation is symmetric in between the configuration space and the momentum space. The Berry (geometric) phase emerges from the vector potential terms in the Lagrangian. Thus just as the adiabatic Berry phase in the momentum space integrates the Berry potential over the orbit, the Berry phase in the configuration space integrates the electromagnetic potential over the orbit giving the magnetic flux.
Namely, the full Berry phase is given by :
$$\phi_B = e^{i \oint e \mathbf{A} \cdot d\mathbf{x} + i \oint e \mathbf{A}_B \cdot d\mathbf{k}} = e^{i \int_{\Sigma} e \mathbf{B} \cdot d\hat{\mathbf{x}} + i \int_{\Sigma_k} \mathbf{\Omega} \cdot d\hat{\mathbf{k}}} $$
Where the line integrals are over an orbit in phase space. The second equality is a consequence of Stokes' theorem; the second surface integral  is the usual Berry phase evaluated in momentum space while the first is the magnetic flux.
