Byers–Yang theorem

An enclosed flux ${\displaystyle \Phi }$ corresponds to a vector potential ${\displaystyle A(r)}$ inside the annulus with a line integral ${\displaystyle \oint _{C}A\cdot dl=\Phi }$ along any path ${\displaystyle C}$ that circulates around once. One can try to eliminate this vector potential by the gauge transformation

${\displaystyle \psi '(\{r_{n}\})=\exp \left({\frac {ie}{\hbar }}\sum _{j}\chi (r_{j})\right)\psi (\{r_{n}\})}$

of the wave function ${\displaystyle \psi (\{r_{n}\})}$ of electrons at positions ${\displaystyle r_{1},r_{2},\ldots }$. The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential ${\displaystyle A'(r)=A(r)+\nabla \chi (r)}$. It is assumed that the electrons experience zero magnetic field ${\displaystyle B(r)=\nabla \times A(r)=0}$ at all points ${\displaystyle r}$ inside the annulus, the field being nonzero only within the opening (where there are no electrons). It is then always possible to find a function ${\displaystyle \chi (r)}$ such that ${\displaystyle A'(r)=0}$ inside the annulus, so one would conclude that the system with enclosed flux ${\displaystyle \Phi }$ is equivalent to a system with zero enclosed flux.

However, for any arbitrary ${\displaystyle \Phi }$ the gauge transformed wave function is no longer single-valued: The phase of ${\displaystyle \psi '}$ changes by

${\displaystyle \delta \phi =(e/\hbar )\oint _{C}\nabla \chi (r)\cdot dl=-(e/\hbar )\oint _{C}A(r)\cdot dl=-2\pi \Phi /\Phi _{0}}$

whenever one of the coordinates ${\displaystyle r_{n}}$ is moved along the ring to its starting point. The requirement of a single-valued wave function therefore restricts the gauge transformation to fluxes ${\displaystyle \Phi }$ that are an integer multiple of ${\displaystyle \Phi _{0}}$. Systems that enclose a flux differing by a multiple of ${\displaystyle h/e}$ are equivalent.

An overview of physical effects governed by the Byers-Yang theorem is given by Yoseph Imry.[3] These include the Aharonov–Bohm effect, persistent current in normal metals, and flux quantization in superconductors.