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 ${\displaystyle \Phi }$ through the opening are periodic in the flux with period ${\displaystyle \Phi _{0}=h/e}$ (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).

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

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