Physics: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 $\mathrm{\Phi }$ through the opening are periodic in the flux with period ${\mathrm{\Phi }}_0=hc/e$ (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 $\mathrm{\Phi }$ corresponds to a vector potential $A(r)$ inside the annulus with a line integral ${\oint }_{C}A\cdot dl=\mathrm{\Phi }$ along any path $C$ that circulates around once. One can try to eliminate this vector potential by the gauge transformation

${\psi }^{\prime }(\{r_n\})=\exp (\frac{ie}{\hslash }\sum _j\chi (r_j))\psi (\{r_n\})$

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

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

$\delta \varphi =(e/\hslash ){\oint }_C\mathrm{\nabla }\chi (r)\cdot dl=-(e/\hslash ){\oint }_{C}A(r)\cdot dl=-2\pi \mathrm{\Phi }/{\mathrm{\Phi }}_0$

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 $\mathrm{\Phi }$ that are an integer multiple of ${\mathrm{\Phi }}_0$. 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.