17.3. Spontaneously broken symmetry in condensed matter physics
203
ˆ
at once that H is invariant under the global U(1) symmetry
ˆ
φ(x) → ˆ
φ
′ (x) = e
−iα ˆ
φ(x),
(17.26)
the generator being the conserved number operator
∫
ˆ
ˆ
N = φ
† ˆ
φ d
3
x
(17.27)
ˆ ˆ
which obeys [N , H] = 0. Our ultimate concern will be with the way this
symmetry is ‘spontaneously broken’ in the superfluid ground state. Naturally,
since this is an Abelian, rather than a non-Abelian, symmetry the physics will
not involve any (hidden) multiplet structure. But the nature of the ‘symmetry
breaking ground state’ in this U(1) case (and in the BCS model of section 17.7)
will serve as a physical model for non-Abelian cases also.
ˆ
We begin by re-writing H in terms of mode creation and annihilation
ˆ
operators in the usual way. We expand φ(x) as a superposition of solutions of
the v = 0 problem, which are plane waves quantized in a large cube of volume
Ω:
1 ∑
ˆ
φ(x) =
Ω
1
a ˆ e
ik·x
(17.28)
2
k
k
where a ˆ |0› = 0, a ˆ
† |0› is a one-particle state, and [a ˆ , a ˆ
† ] = δ
k
k
k k
′
k,k
′ , with
all other commutators vanishing. We impose periodic boundary conditions
at the cube faces, and the free particle energies are ∈ k = k
2 /2m. Inserting
(17.28) into (17.25) leads (problem 17.1) to
∑
1 ∑
ˆ
H =
∈ k a ˆ
† a +
v ¯(
k
k k
|k 1 −
′
1 |)ˆ a
† a ˆ
† a ˆ
k k k
′ a ˆ ′ Δ(k 1 + k 2 k
′
k
′ )
2Ω
2 k
1
1
2
1
k
Δ
−
− 2
(17.29)
where the sum is over all momenta k 1 , k 2 , k
′
1 , k
′
2 subject to the conservation
law imposed by the Δ function:
Δ(k) = 1
if k = 0
(17.30)
= 0
if k = 0.
(17.31)
The interaction term in (17.29) is easily visualized as in figure 17.1. A pair
of particles in states k
′
1 , k
′
2 is scattered (conserving momentum) to a pair in
states k 1 , k 2 via the Fourier transform of v:
∫
v ¯(|k|
k
) = v(r)e
−i ·r d
3
r.
(17.32)
Now, below the superfluid transition temperature T S , we know that in the
limit as v → 0 the ground state has all the particles ‘condensed’ into the lowest
energy state, which has k = 0. Thus the ground state will be proportional to
|N, 0› = (a ˆ
†
0 )
N
|0›.
(17.33)
/
203
ˆ
at once that H is invariant under the global U(1) symmetry
ˆ
φ(x) → ˆ
φ
′ (x) = e
−iα ˆ
φ(x),
(17.26)
the generator being the conserved number operator
∫
ˆ
ˆ
N = φ
† ˆ
φ d
3
x
(17.27)
ˆ ˆ
which obeys [N , H] = 0. Our ultimate concern will be with the way this
symmetry is ‘spontaneously broken’ in the superfluid ground state. Naturally,
since this is an Abelian, rather than a non-Abelian, symmetry the physics will
not involve any (hidden) multiplet structure. But the nature of the ‘symmetry
breaking ground state’ in this U(1) case (and in the BCS model of section 17.7)
will serve as a physical model for non-Abelian cases also.
ˆ
We begin by re-writing H in terms of mode creation and annihilation
ˆ
operators in the usual way. We expand φ(x) as a superposition of solutions of
the v = 0 problem, which are plane waves quantized in a large cube of volume
Ω:
1 ∑
ˆ
φ(x) =
Ω
1
a ˆ e
ik·x
(17.28)
2
k
k
where a ˆ |0› = 0, a ˆ
† |0› is a one-particle state, and [a ˆ , a ˆ
† ] = δ
k
k
k k
′
k,k
′ , with
all other commutators vanishing. We impose periodic boundary conditions
at the cube faces, and the free particle energies are ∈ k = k
2 /2m. Inserting
(17.28) into (17.25) leads (problem 17.1) to
∑
1 ∑
ˆ
H =
∈ k a ˆ
† a +
v ¯(
k
k k
|k 1 −
′
1 |)ˆ a
† a ˆ
† a ˆ
k k k
′ a ˆ ′ Δ(k 1 + k 2 k
′
k
′ )
2Ω
2 k
1
1
2
1
k
Δ
−
− 2
(17.29)
where the sum is over all momenta k 1 , k 2 , k
′
1 , k
′
2 subject to the conservation
law imposed by the Δ function:
Δ(k) = 1
if k = 0
(17.30)
= 0
if k = 0.
(17.31)
The interaction term in (17.29) is easily visualized as in figure 17.1. A pair
of particles in states k
′
1 , k
′
2 is scattered (conserving momentum) to a pair in
states k 1 , k 2 via the Fourier transform of v:
∫
v ¯(|k|
k
) = v(r)e
−i ·r d
3
r.
(17.32)
Now, below the superfluid transition temperature T S , we know that in the
limit as v → 0 the ground state has all the particles ‘condensed’ into the lowest
energy state, which has k = 0. Thus the ground state will be proportional to
|N, 0› = (a ˆ
†
0 )
N
|0›.
(17.33)
/