In quantum mechanics, the Byers-Yang theorem states that all physical properties of a doubly connected system (an annulus) enclosing a magnetic flux
Φ
through the opening are periodic in the flux with period
Φ
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
Φ
corresponds to a vector potential
A
(
r
)
inside the annulus with a line integral
∮
C
A
⋅
d
l
=
Φ
along any path
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
}
)
of the wave function
ψ
(
{
r
n
}
)
of electrons at positions
r
1
,
r
2
,
…
. 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
)
. It is assumed that the electrons experience zero magnetic field
B
(
r
)
=
∇
×
A
(
r
)
=
0
at all points
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
)
such that
A
′
(
r
)
=
0
inside the annulus, so one would conclude that the system with enclosed flux
Φ
is equivalent to a system with zero enclosed flux.
However, for any arbitrary
Φ
the gauge transformed wave function is no longer single-valued: The phase of
ψ
′
changes by
δ
ϕ
=
(
e
/
ℏ
)
∮
C
∇
χ
(
r
)
⋅
d
l
=
2
π
Φ
/
Φ
0
whenever one of the coordinates
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
that are an integer multiple of
Φ
0
. Systems that enclose a flux differing by a multiple of
h
/
e
are equivalent.
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.
