Chern number

An integer topological invariant obtained by integrating Berry curvature over a closed parameter space. In quantum systems, it can determine quantized responses such as the Hall conductance.

Berry curvature: The curvature of the geometric phase acquired by a quantum state as its parameters vary. Its integral over a closed parameter space gives the Chern number.
Integer quantum Hall effect: A two-dimensional electron system exhibits Hall conductance quantized in integer multiples of e²/h. Filled bands' Chern numbers determine the quantized Hall conductance.
Topology: The mathematical study of properties preserved under continuous deformation. Chern numbers classify global features that smooth changes cannot remove.
Zak phase: The Berry phase accumulated along a one-dimensional path through a crystal's momentum space. It is a one-dimensional geometric phase, unlike the two-dimensional integral defining a band Chern number.
Berry connection: A gauge-dependent vector potential whose curvature is Berry curvature. It supplies the local gauge potentials used to compute Berry curvature.
Chern insulator: A two-dimensional insulating phase with nonzero band Chern number and chiral edge states. It realizes a nonzero Chern number without an external magnetic field.
Complex line bundle: A space that assigns a one-dimensional complex vector space to each point of a base space. An isolated quantum band forms such a bundle over its momentum-space parameter space.
Winding number: An integer counting how many times a map wraps one circle around another. It is another integer topological invariant, often defined from a lower-dimensional map.
Brillouin zone: The primitive cell of reciprocal space for a periodic crystal. In two-dimensional band systems, it is commonly the closed parameter space over which curvature is integrated.
TKNN invariant: The band-topological integer linking Bloch-state geometry to quantized Hall conductance. It established the Chern number as the invariant behind the integer quantum Hall effect.

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