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. 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. Such phases have come to be known as Berry phases.
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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
where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with
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
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
Without changing the physics, we can make a gauge transformation
to a new set of states that differ from the original ones only by an
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
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
For a closed path
If the surface is a closed manifold, the boundary term vanishes, but the indeterminacy of the boundary term modulo
Finally, note that the Berry curvature can also be written, using perturbation theory, 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
Now consider the
The Berry curvature per solid angle is given by
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
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.