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),[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 \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.[3] These include the Aharonov-Bohm effect, persistent current in normal metals, and flux quantization in superconductors.

Notes

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