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Byers–Yang theorem

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In quantum mechanics, the Byers–Yang theorem states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux through the opening are periodic in the flux with period (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

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An enclosed flux corresponds to a vector potential inside the annulus with a line integral along any path that circulates around once. One can try to eliminate this vector potential by the gauge transformation

of the wave function of electrons at positions . The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential . It is assumed that the electrons experience zero magnetic field at all points 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 such that inside the annulus, so one would conclude that the system with enclosed flux is equivalent to a system with zero enclosed flux.

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

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

Applications

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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.

References

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  1. ^ 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.
  2. ^ 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.
  3. ^ Imry, Y. (1997). Introduction to Mesoscopic Physics. Oxford University Press. ISBN 0-19-510167-7.