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 [math]\displaystyle{ \Phi }[/math] through the opening are periodic in the flux with period [math]\displaystyle{ \Phi_0=hc/e }[/math] (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 [math]\displaystyle{ \Phi }[/math] corresponds to a vector potential [math]\displaystyle{ A(r) }[/math] inside the annulus with a line integral [math]\displaystyle{ \oint_C A\cdot dl=\Phi }[/math] along any path [math]\displaystyle{ C }[/math] that circulates around once. One can try to eliminate this vector potential by the gauge transformation
- [math]\displaystyle{ \psi'(\{r_n\})=\exp\left(\frac{ie}{\hbar}\sum_j\chi(r_j)\right)\psi(\{r_n\}) }[/math]
of the wave function [math]\displaystyle{ \psi(\{r_n\}) }[/math] of electrons at positions [math]\displaystyle{ 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]\displaystyle{ A'(r)=A(r)+\nabla\chi(r) }[/math]. It is assumed that the electrons experience zero magnetic field [math]\displaystyle{ B(r)=\nabla\times A(r)=0 }[/math] at all points [math]\displaystyle{ 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]\displaystyle{ \chi(r) }[/math] such that [math]\displaystyle{ A'(r)=0 }[/math] inside the annulus, so one would conclude that the system with enclosed flux [math]\displaystyle{ \Phi }[/math] is equivalent to a system with zero enclosed flux.
However, for any arbitrary [math]\displaystyle{ \Phi }[/math] the gauge transformed wave function is no longer single-valued: The phase of [math]\displaystyle{ \psi' }[/math] changes by
- [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \Phi }[/math] that are an integer multiple of [math]\displaystyle{ \Phi_0 }[/math]. Systems that enclose a flux differing by a multiple of [math]\displaystyle{ h/e }[/math] 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.
References
- ↑ Byers, N.; Yang, C. N. (1961). "Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders". Physical Review Letters 7 (2): 46–49. doi:10.1103/PhysRevLett.7.46. Bibcode: 1961PhRvL...7...46B.
- ↑ Bloch, F. (1970). "Josephson Effect in a Superconducting Ring". Physical Review B 2 (1): 109–121. doi:10.1103/PhysRevB.2.109. Bibcode: 1970PhRvB...2..109B.
- ↑ Imry, Y. (1997). Introduction to Mesoscopic Physics. Oxford University Press. ISBN 0-19-510167-7.
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