Byers-Yang theorem

In quantum mechanics, the Byers-Yang theorem states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux $\Phi$ through the opening are periodic in the flux with period $\Phi_0=h/e$ (the magnetic flux quantum). The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961), and further developed by Felix Bloch (1970).

Proof

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

$\psi'(\{r_n\})=\exp\left(\frac{ie}{\hbar}\sum_j\chi(r_j)\right)\psi(\{r_n\})$

of the wave function $\psi(\{r_n\})$ of electrons at positions $r_1,r_2,\ldots$. The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential $A'(r)=A(r)+\nabla\chi(r)$. It is assumed that the electrons experience zero magnetic field $B(r)=\nabla\times A(r)=0$ at all points $r$ 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 $\chi(r)$ such that $A'(r)=0$ inside the annulus, so one would conclude that the system with enclosed flux $\Phi$ is equivalent to a system with zero enclosed flux.

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

$\delta\phi=(e/\hbar)\oint_C\nabla\chi(r)\cdot dl=2\pi\Phi/\Phi_0$

whenever one of the coordinates $r_n$ is moved along the ring to its starting point. The requirement of a single-valued wave function therefore restricts the gauge transformation to fluxes $\Phi$ that are an integer multiple of $\Phi_0$. Systems that enclose a flux differing by a multiple of $h/e$ 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.