Berry connection and curvature

In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
Berry phase and cyclic adiabatic evolution.
In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian formula_1 depends on a (vector) parameter formula_2 that varies with time formula_3. If the formula_4'th eigenvalue formula_5 remains non-degenerate everywhere along the path and the variation with time "t" is sufficiently slow, then a system initially in the eigenstate 
formula_6 will remain in an instantaneous eigenstate formula_7 of the Hamiltonian formula_8, up to a phase, throughout the process. Regarding the phase, the state at time "t" can be written as
where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with formula_10 being the Berry phase. From the requirement that formula_11 satisfy the time-dependent Schrödinger equation, it can be shown that
indicating that the Berry phase only depends on the path in the parameter space, not on the rate at which the path is traversed.
In the case of a cyclic evolution around a closed path formula_13 such that formula_14, the closed-path Berry phase is
An example of physical system where an electron moves along a closed path is cyclotron motion (details are given in the page of Berry phase). Berry phase must be considered to obtain the correct quantization condition.
Gauge transformation.
A gauge transformation can be performed
to a new set of states that differ from the original ones only by an formula_2-dependent phase factor. This modifies the open-path Berry phase to be formula_18. For a closed path, continuity requires that formula_19 (formula_20 an integer), and it follows that formula_10 is invariant, modulo formula_22, under an arbitrary gauge transformation.
Berry connection.
The closed-path Berry phase defined above can be expressed as
where 
is a vector-valued function known as the Berry connection (or Berry potential). The Berry connection is gauge-dependent, transforming as
formula_25. Hence the local Berry connection formula_26 can never be physically observable. However, its integral along a closed path, the Berry phase formula_10, is gauge-invariant up to an integer multiple of formula_22. Thus, formula_29 is absolutely gauge-invariant, and may be related to physical observables.
Berry curvature.
The Berry curvature is an anti-symmetric second-rank tensor derived from the Berry connection via
In a three-dimensional parameter space the Berry curvature can be written in the pseudovector form
The tensor and pseudovector forms of the Berry curvature are related to each other through the Levi-Civita antisymmetric tensor as
formula_32. In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.
For a closed path formula_13 that forms the boundary of a surface formula_34, the closed-path Berry phase can be rewritten using Stokes' theorem as
If the surface is a closed manifold, the boundary term vanishes, but the indeterminacy of the boundary term modulo formula_22 manifests itself in the Chern theorem, which states that the integral of the Berry curvature over a closed manifold is quantized in units of formula_22. This number is the so-called Chern number, and is essential for understanding various quantization effects.
Finally, note that the Berry curvature can also be written as a sum over all other eigenstates in the form
Example: Spinor in a magnetic field.
The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as
where formula_40 denote the Pauli matrices, formula_41 is the magnetic moment, and B is the magnetic field. In three dimensions, the eigenstates have energies formula_42 and their eigenvectors are
Now consider the formula_44 state. Its Berry connection can be computed as
formula_45
formula_46,
and the Berry curvature is
formula_47
If we choose a new gauge by multiplying formula_44 by formula_49, the Berry connections are
formula_50 and formula_51, while the Berry curvature remains the same. This is consistent with the conclusion that the Berry connection is gauge-dependent while the Berry curvature is not.
The Berry curvature per solid angle is given by formula_52. In this case, the Berry phase corresponding to any given path on the unit sphere formula_53 in magnetic-field space is just half the solid angle subtended by the path.
The integral of the Berry curvature over the whole sphere is therefore exactly formula_22, so that the Chern number is unity, consistent with the Chern theorem.
Applications in crystals.
The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids and in the theory of the quantum Hall effect.
The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form
where formula_4 is a band index, formula_57 is a wavevector in the reciprocal-space (Brillouin zone), and formula_58 is a periodic function of formula_59. Then, letting formula_57 play the role of the parameter formula_2, one can define Berry phases, connections, and curvatures in the reciprocal space. For example, the Berry connection in reciprocal space is
Because the Bloch theorem also implies that the reciprocal space itself is closed, with the Brillouin zone having the topology of a 3-torus in three dimensions, the requirements of integrating over a closed loop or manifold can easily be satisfied. In this way, such properties as the electric polarization, orbital magnetization, anomalous Hall conductivity, and orbital magnetoelectric coupling can be expressed in terms of Berry phases, connections, and curvatures.