Physics:Byers–Yang theorem

Short description: Theorem in quantum mechanics

In quantum mechanics, the Byers–Yang theorem states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux $\mathrm{\Phi }$ through the opening are periodic in the flux with period ${\mathrm{\Phi }}_0=hc/e$ (the magnetic flux quantum). The theorem was first stated and proven by Nina Byers and Chen-Ning Yang (1961), and further developed by Felix Bloch (1970).

Proof

An enclosed flux $\mathrm{\Phi }$ corresponds to a vector potential $\mathbf{𝐀}(\mathbf{𝐫})$ inside the annulus with a line integral ${\oint }_C\mathbf{𝐀}\cdot \mathrm{d}\mathbf{𝐥}=\mathrm{\Phi }$ along any path $C$ that circulates around once. One can try to eliminate this vector potential by the gauge transformation

${\psi }^{\prime }(\{{\mathbf{𝐫}}_n\})=\exp (\frac{ie}{\hslash }\sum _j\chi ({\mathbf{𝐫}}_j))\psi (\{{\mathbf{𝐫}}_n\})$

of the wave function $\psi (\{{\mathbf{𝐫}}_n\})$ of electrons at positions ${\mathbf{𝐫}}_1,{\mathbf{𝐫}}_2,\dots$. The gauge-transformed wave function satisfies the same Schrödinger equation as the original wave function, but with a different magnetic vector potential ${\mathbf{𝐀}}^{\prime }(\mathbf{𝐫})=\mathbf{𝐀}(\mathbf{𝐫})+\mathrm{\nabla }\chi (\mathbf{𝐫})$. It is assumed that the electrons experience zero magnetic field $\mathbf{𝐁}(\mathbf{𝐫})=\mathrm{\nabla }\times \mathbf{𝐀}(\mathbf{𝐫})=0$ at all points $\mathbf{𝐫}$ 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 $\chi (\mathbf{𝐫})$ such that ${\mathbf{𝐀}}^{\prime }(\mathbf{𝐫})=0$ inside the annulus, so one would conclude that the system with enclosed flux $\mathrm{\Phi }$ is equivalent to a system with zero enclosed flux.

However, for any arbitrary $\mathrm{\Phi }$ the gauge transformed wave function is no longer single-valued: The phase of ${\psi }^{\prime }$ changes by

$\delta \varphi =\frac{e}{\hslash }{\oint }_C\mathrm{\nabla }\chi (\mathbf{𝐫})\cdot \mathrm{d}\mathbf{𝐥}=-\frac{e}{\hslash }{\oint }_C\mathbf{𝐀}(\mathbf{𝐫})\cdot \mathrm{d}\mathbf{𝐥}=-2\pi \frac{\mathrm{\Phi }}{{\mathrm{\Phi }}_0}$

whenever one of the coordinates ${\mathbf{𝐫}}_n$ is moved along the ring to its starting point. The requirement of a single-valued wave function therefore restricts the gauge transformation to fluxes $\mathrm{\Phi }$ that are an integer multiple of ${\mathrm{\Phi }}_0$. 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. These include the Aharonov–Bohm effect, persistent current in normal metals, and flux quantization in superconductors.