The concept was first introduced by S. Pancharatnam[1] as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984[2] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution.
The quantum adiabatic theorem applies to a system whose Hamiltonian
remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the normalized eigenstate
where the second exponential term is the "dynamic phase factor."
satisfies the time-dependent Schrödinger equation, it can be shown that
In the case of a cyclic evolution around a closed path
An example of physical systems 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.
The closed-path Berry phase defined above can be expressed as
is absolutely gauge-invariant, and may be related to physical observables.
In a three-dimensional parameter space the Berry curvature can be written in the pseudovector form
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.
, the closed-path Berry phase can be rewritten using Stokes' theorem as
manifests itself in the Chern theorem, which states that the integral of the Berry curvature over a closed manifold is quantized in units of
, the Berry curvature can also be written as a summation over all the other eigenstates in the form
That is, the Berry curvature can be viewed as the result of the residual interaction of those projected-out eigenstates.
This equation also offers the advantage that no differentiation on the eigenstates is involved, and thus it can be computed under any gauge choice.
The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as[3]
In this case, the Berry phase corresponding to any given path on the unit sphere
in magnetic-field space is just half the solid angle subtended by the path.
The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids[5] and in the theory of the quantum Hall effect.
[6] The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form
Due to translational symmetry, the momentum operator
[5] Thus, one can define Berry phases, connections, and curvatures in the reciprocal space.
For example, in an N-band system, the Berry connection of the nth band in reciprocal space is
In the system, the Berry curvature of the nth band
In a 2D crystal, the Berry curvature only has the component out of the plane and behaves as a pseudoscalar.
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.