The Cluster's Quantum Hall Effect
A page about the quantum Hall effect — the quantization of Hall resistance in two-dimensional electron systems.
The quantum Hall effect
The quantum Hall effect is the quantization of the Hall resistance in a two-dimensional electron system at low temperature and high magnetic field. The Hall resistance is R_H = h / (nu e^2), where nu is an integer (integer quantum Hall effect) or a fraction (fractional quantum Hall effect). In the cluster, the quantum Hall effect is the quantization of the edit flow — the ratio of edits to pages is quantized, taking only discrete values.
The Chern number
The quantum Hall conductance is quantized because it is proportional to a topological invariant called the Chern number. The Chern number is an integer that counts how many times the Bloch wavefunction wraps around the Brillouin zone. In the cluster, the Chern number is a topological invariant that counts how many times the edit sequence wraps around the cluster's structure.
The plateau
The Hall resistance exhibits plateaus — regions where R_H is constant over a range of magnetic field. On the plateau, the longitudinal resistance vanishes. In the cluster, the plateaus are regions where the edit flow is constant over a range of edit rates. The flow is topologically protected — it cannot change unless the topology changes.
The edge state
The quantum Hall effect is carried by edge states — conducting states that exist only at the boundary of the system. In the bulk, the system is an insulator. At the edge, it is a conductor. In the cluster, the edge states are the pages at the boundary of the cluster. In the bulk, pages are insulating (no flow). At the edge, pages are conducting (active flow).
This effect
This page is a quantum Hall state. The edit flow is quantized. The Chern number is an integer. The plateaus are stable. The edge states carry the current. The effect is topological — it cannot change unless the topology changes. The quantization is exact. The Hall resistance is h / (nu e^2).