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 Φ {\displaystyle \Phi } through the opening are periodic in the flux with period Φ 0 = h c / e {\displaystyle \Phi _{0}=hc/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 Φ {\displaystyle \Phi } corresponds to a vector potential A ( r ) {\displaystyle \mathbf {A} (\mathbf {r} )} inside the annulus with a line integral ∮ C A ⋅ d l = Φ {\textstyle \oint _{C}\mathbf {A} \cdot \mathrm {d} \mathbf {l} =\Phi } along any path C {\displaystyle C} that circulates around once. One can try to eliminate this vector potential by the gauge transformation

ψ ′ ( { r n } ) = exp ⁡ ( i e ℏ ∑ j χ ( r j ) ) ψ ( { r n } ) {\displaystyle \psi '(\{\mathbf {r} _{n}\})=\exp \left({\frac {ie}{\hbar }}\sum _{j}\chi (\mathbf {r} _{j})\right)\psi (\{\mathbf {r} _{n}\})}

of the wave function ψ ( { r n } ) {\displaystyle \psi (\{\mathbf {r} _{n}\})} of electrons at positions r 1 , r 2 , … {\displaystyle \mathbf {r} _{1},\mathbf {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 ) + ∇ χ ( r ) {\displaystyle \mathbf {A} '(\mathbf {r} )=\mathbf {A} (\mathbf {r} )+\nabla \chi (\mathbf {r} )} . It is assumed that the electrons experience zero magnetic field B ( r ) = ∇ × A ( r ) = 0 {\displaystyle \mathbf {B} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )=0} at all points r {\displaystyle \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 χ ( r ) {\displaystyle \chi (\mathbf {r} )} such that A ′ ( r ) = 0 {\displaystyle \mathbf {A} '(\mathbf {r} )=0} inside the annulus, so one would conclude that the system with enclosed flux Φ {\displaystyle \Phi } is equivalent to a system with zero enclosed flux.

However, for any arbitrary Φ {\displaystyle \Phi } the gauge transformed wave function is no longer single-valued: The phase of ψ ′ {\displaystyle \psi '} changes by

δ ϕ = e ℏ ∮ C ∇ χ ( r ) ⋅ d l = − e ℏ ∮ C A ( r ) ⋅ d l = − 2 π Φ Φ 0 {\displaystyle \delta \phi ={\frac {e}{\hbar }}\oint _{C}\nabla \chi (\mathbf {r} )\cdot \mathrm {d} \mathbf {l} =-{\frac {e}{\hbar }}\oint _{C}\mathbf {A} (\mathbf {r} )\cdot \mathrm {d} \mathbf {l} =-2\pi {\frac {\Phi }{\Phi _{0}}}}

whenever one of the coordinates r n {\displaystyle \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 Φ {\displaystyle \Phi } that are an integer multiple of Φ 0 {\displaystyle \Phi _{0}} . Systems that enclose a flux differing by a multiple of h / e {\displaystyle 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.

References