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 <math>\Phi</math> through the opening are periodic in the flux with period <math>\Phi_0=hc/e</math> (the magnetic flux quantum). The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961),<ref>Byers, N.; Yang, C. N. (1961). "Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders". Physical Review Letters. 7 (2): 46–49. Bibcode:1961PhRvL...7...46B. doi:10.1103/PhysRevLett.7.46.</ref> and further developed by Felix Bloch (1970).<ref>Bloch, F. (1970). "Josephson Effect in a Superconducting Ring". Physical Review B. 2 (1): 109–121. Bibcode:1970PhRvB...2..109B. doi:10.1103/PhysRevB.2.109.</ref>

Proof

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

<math>\psi'(\{r_n\})=\exp\left(\frac{ie}{\hbar}\sum_j\chi(r_j)\right)\psi(\{r_n\})</math>

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

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

<math>\delta\phi=(e/\hbar)\oint_C\nabla\chi(r)\cdot dl=-(e/\hbar)\oint_C A(r)\cdot dl=-2\pi\Phi/\Phi_0</math>

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

Applications

An overview of physical effects governed by the Byers–Yang theorem is given by Yoseph Imry.<ref>Imry, Y. (1997). Introduction to Mesoscopic Physics. Oxford University Press. ISBN 0-19-510167-7.</ref> These include the Aharonov–Bohm effect, persistent current in normal metals, and flux quantization in superconductors.