Byers-Yang theorem


In quantum mechanics, the Byers-Yang theorem states that all physical properties of a doubly connected system enclosing a magnetic flux through the opening are periodic in the flux with period . The theorem was first stated and proven by Nina Byers and Chen-Ning Yang, and further developed by Felix Bloch.

Proof

An enclosed flux corresponds to a vector potential inside the annulus with a line integral along any path that circulates around once. One can try to eliminate this vector potential by the gauge transformation
of the wave function of electrons at positions. The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential. It is assumed that the electrons experience zero magnetic field at all points inside the annulus, the field being nonzero only within the opening. It is then always possible to find a function such that inside the annulus, so one would conclude that the system with enclosed flux is equivalent to a system with zero enclosed flux.
However, for any arbitrary the gauge transformed wave function is no longer single-valued: The phase of changes by
whenever one of the coordinates is moved along the ring to its starting point. The requirement of a single-valued wave function therefore restricts the gauge transformation to fluxes that are an integer multiple of. Systems that enclose a flux differing by a multiple of are equivalent.

Applications

An overview of physical effects governed by the Byers-Yang theorem is given by Yoseph Imry. These include the
Aharonov-Bohm effect, persistent current in normal metals, and flux quantization in superconductors.