Byers–Yang theorem explained

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=hc/e} (the magnetic flux quantum). The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961),^[1] and further developed by Felix Bloch (1970).^[2]

Proof

An enclosed flux

\Phi corresponds to a vector potential A(r) inside the annulus with a line integral $\backslash oint\_C\; A\backslash cdot\; dl=\backslash Phi$ 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( (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 x 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) ⋅} dl =- (e/\hbar)\oint _{C} A(r) ⋅ 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.^[3] These include theAharonov–Bohm effect, persistent current in normal metals, and flux quantization in superconductors.

Notes and References

1. Byers . N. . Nina Byers . Yang . C. N. . Yang Chen-Ning . Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders . . 1961. 7. 2. 46–49 . 10.1103/PhysRevLett.7.46 . 1961PhRvL...7...46B .
2. Bloch. F.. 1970. Josephson Effect in a Superconducting Ring . Physical Review B. 2. 1 . 109–121. 10.1103/PhysRevB.2.109 . 1970PhRvB...2..109B .
3. Book: Imry, Y.. Introduction to Mesoscopic Physics. Oxford University Press. 1997. 0-19-510167-7 .