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Since Michael Berry published the paper about the so-called Berry phase in 1984 , it has become a powerful unifying concept, especially valuable in specializing physical science.

This article discusses the Berry phase of a quantum system in an adiabatic evolution, and introduces Berry connection and Berry curvature, which can be viewed as the vector potential and local description of the Berry phase, repectively.

Berry phase and cyclic adiabatic evolution
Quantum mechanically, the Berry phase is considered in a cyclic adiabatic evolution. According to the quantum adiabatic theorem, a system initially in one of its eigenstates $$\, |n(\mathbf R(0))\rangle $$ will stay as an instantaneous eigenstate of the Hamiltonian $$\, H(\mathbf R(t))$$ throughout the process, where $$\, \mathbf R(t)$$ denotes a parameter in the Hamiltonian. Therefore the only degree of freedom is the phase of the quantum state. The state at time t can be written as

$$ where the second exponential term is the dynamic factor in the state, and $$\varepsilon_n$$ represents the nth eigenvalue. The first exponential term is the geometric factor, with the Berry phase $$\gamma_n$$ relevant to geometric paths in the parameter space. It can be derived that
 * \Psi_n(t)\rangle =e^{i\gamma_n(t)}\exp\left[ -{i\over\hbar}\int_0 ^t dt'\varepsilon_n(\mathbf R(t'))\right]| n(\mathbf R(0))\rangle,

\gamma_n(t)=i\int_\mathcal{C}^t dt'\langle n(\mathbf R(t'))|{d\over dt'}|n(\mathbf R(t'))\rangle=i\int_\mathcal{C}^\mathbf R d\mathbf R\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle, $$ indicating that the Berry phase is only dependent on the path in the parameter space, but independent of how fast it evolves.

Berry connection
Following the Berry phase defined above, we can express it as $$\gamma_n=\int_\mathcal{C} d\mathbf R\cdot \mathcal{A}_n(\mathbf R)$$, where the vector-valued function in the parameter space $$\mathcal A_n$$ is the Berry connection. Write it explicitly,

\mathcal{A}_n(\mathbf R)=i\langle n(\mathbf R)|\nabla_{\mathbf R}|n(\mathbf R)\rangle. $$ Note that the Berry connection is gauge dependent. For instance, we can make a gauge transform $$|\tilde n(\mathbf R)\rangle=e^{-i\beta(\mathbf R)}|n(\mathbf R)\rangle$$, as a consequence, $$\tilde{\mathcal{A}}_n(\mathbf R)=\nabla_{\mathbf R}\beta(\mathbf R)+\mathcal{A}_n (\mathbf R)$$. Hence the Berry connection at one point or along an open path cannot be observed, because it changes with different gauges. However, its integral along a closed path is gauge invariant, though it can be changed by an integer multiple of $$2\pi$$. This leads to the well-defined Berry phase along a closed loop in the parameter space, since the Berry phase can be a number modulo $$2\pi$$ and that would not affect the physics.

Berry curvature
The Berry phase can also be expressed in terms of a local geometrical quantity, Berry curvature, over the surface suspending the closed loop in the parameter space. The Berry curvature is an anti-symmetric gauge-field tensor derived from the Berry connection :

\Omega_{\mu\nu}^n (\mathbf R)={\partial\over\partial R^\mu}\mathcal{A}_\nu^n(\mathbf R)-{\partial\over\partial R^\nu}\mathcal{A}_\mu^n(\mathbf R). $$ In three-dimensional parameter space the Berry curvature can be written in a vector form

\mathbf\Omega_n(\mathbf R)=\nabla_{\mathbf R} \times\mathcal{A}_n(\mathbf R), $$ The Berry curvature tensor and vector are related to each other by the Levi-Civita antisymmetric tensor $$\Omega_{\mu\nu}^n=\epsilon_{\mu\nu\xi}(\mathbf\Omega_n)_{\xi}$$. Then we can rewrite the Berry phase in an integral over a surface,

\gamma_n=\int_\mathcal{S} d\mathbf S\cdot\mathbf\Omega_n (\mathbf R). $$

Compared to the Berry connection which is physical only in a closed path, the Berry curvature provides a gauge-invariant and local description of the geometric properties of the parameter space and is proved to be an essential physical quantity for all kinds of electronic properties . In general, the Berry curvature integrated over a closed manifold is quantized in the units of $$2\pi$$ and represents the net number of monopoles in side. This number is the so-called Chern number and it is associated with various quantization effects.

In addition, with the help of Hellmann-Feynman theorem, the Berry curvature can be also written as a summation over the eigenstates:

\Omega_{\mu\nu}^n(\mathbf R)=i\sum_{n\neq n'}{\langle n|\partial H/\partial R^\mu |n'\rangle\langle n'|\partial H/\partial R^\nu | n\rangle-(\nu\leftrightarrow\mu)\over(\varepsilon_n-\varepsilon_{n'})^2}, $$ which, considering the adiabatic approximation is actually a projection operation on an eigenstate, indicates that the Berry curvature can be regarded as the result of the "residual" interaction of those projected-out energy levels.

Example: 1/2 spinor
The Hamiltonian of a 1/2-spinor in the magnetic field can be written as

H=\mu\mathbf\sigma\cdot\mathbf B, $$ where $$\mathbf\sigma$$ denote the Pauli matrices, $$\mu$$ is the magnetic moment, and B is the magnetic field. In three dimensions, the eigenstates can be obtained, with energies $$\pm\mu B$$:

\begin{pmatrix} \sin{\theta\over 2}e^{-i\phi}\\ -\cos{\theta/2} \end{pmatrix}, \begin{pmatrix} \cos{\theta\over 2}e^{-i\phi}\\ \sin{\theta/2} \end{pmatrix}. $$ Now consider the $$|u_-\rangle$$ state, the Berry connections can be computed as $$\mathcal{A}_\theta=\langle u|i\partial_\theta u\rangle=0 $$, $$ \mathcal{A}_\phi=\langle u|i\partial_\phi u\rangle=\sin^2{\theta\over 2} $$, and the Berry curvature is $$ \Omega_{\theta\phi}=\partial_\theta\mathcal{A}_\phi-\partial_\phi\mathcal A_\theta={1\over 2}\sin\theta $$. We can choose another gauge by multiplying $$|u_-\rangle$$ by $$e^{i\phi}$$. Similarly the Berry connections are $$\mathcal{A}_\theta=0$$ and $$\mathcal{A}_\phi=-\cos^2{\theta\over 2}$$, with Berry curvature the same. This is coincident with the conclusion that the Berry connection is gauge dependent while the Berry curvature is not.
 * u_-\rangle=
 * u_+\rangle=

The Berry curvature per solid angle is given as $$\overline{\Omega}_{\theta\phi}=\Omega_{\theta\phi}/\sin\theta=1/2$$. In this case, Berry phases correlated to any paths on the $$\mathcal S^2$$ surface of parameters $$\theta,\phi$$ can be calculated as the path's solid angle multiplied by 1/2. Of interest is to integrate the Berry curvature over the whole sphere, containing the monopole at B=0, and the result is $$2\pi$$, namely $$2\pi$$ times a Chern number as we discussed above.

Application in crystals
The Berry phase plays a critical role in modern investigations of electronic properties in the crystal, such as electric polarization and orbital magnetization. . Because of the periodic of crystalline potential, it is convenient to study the crystal in the wavevector space (or reciprocal space) in which the Bloch theorem shows that eigenstates of a Hamiltonian with the periodic potential are of the form

\psi_{nk}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{nk}(\mathbf r), $$ where $$\psi_{nk}$$ is the Bloch state, periodic in k space, with the subscribe n showing that it is the nth eigenstate of the Hamiltonian, relating to the nth energy band of the crystal. Usually, physicists are prone to use $$u_{nk}(\mathbf r)=e^{-i\mathbf k\cdot\mathbf r}\psi_{nk}(\mathbf r)$$ to study the Berry curvature and Berry connection in crystals, with their wavevectors being the parameters, since its periodicity in cells promises the well-behaved results at large r in computation. The periodicity of the band structure in k space is well displayed by the Brillouin zone(BZ), and the requirements of integrating over a closed loop and manifold can be considered as a line integral, and a surface integral over the 1st BZ, respectively.