202
17. Spontaneously Broken Global Symmetry
feature that for low q (long wavelength) their frequency ω tends to zero with
q (actually ω ∝ q
2 ). In this respect, therefore, they behave like massless particles when quantized – and this is another feature we should expect when a
symmetry is spontaneously broken.
The ferromagnet gives us one more useful insight. We have been assuming
that one particular ground state (e.g. the one with S z = N/2) has been somehow ‘chosen’. But what does the choosing? The answer to this is clear enough
ˆ
in the (perfectly r ealistic) case in which the Hamiltonian H S is supplemented
ˆ
by a term −gμB i S iz , representing the effect of an applied field B directed
along the z-axis. This term will indeed ensure that the ground state is unique,
and has S z = N/2. Consider now the two limits B → 0 and N → ∞, both
at finite temperature. When B → 0 at finite N , the N + 1 different S z eigenstates become degenerate, and we have an ensemble in which each enters with
an equal weight; there is therefore no loss of symmetry, even as N → ∞ (but
only after B → 0). On the other hand, if N → ∞ at finite B = 0, the single
state with S z = N/2 will be selected out as the unique ground state and this
asymmetric situation will persist even in the limit B → 0. In a (classical)
mean field theory approximation we suppose that an ‘internal field’ is ‘spontaneously generated’, which is aligned with the external B and survives even
as B → 0, thus ‘spontaneously’ breaking the symmetry.
The ferromagnet therefore provides an easily pictured system exhibiting
many of the features associated with spontaneous symmetry breaking; most
importantly, it strongly suggests that what is really characteristic about the
phenonenon is that it entails ‘spontaneous ordering’.
3 Generally such ordering
occurs below some characteristic ‘critical temperature’, T C . The field which
develops a non-zero equilibrium value below T C is called an ‘order parameter’. This concept forms the basis of Landau’s theory of second-order phase
transitions (see for example chapter XIV of Landau and Lifshitz 1980).
We now turn to an example much more closely analogous to the particle
physics applications: the superfluid.
17.3.2 The Bogoliubov superfluid
Consider the non-relativistic Hamiltonian (in the Schro ¨dinger picture)
∫
1
ˆ
H =
d
3
x
†
m
∇ ˆ
φ
2
· ∇ ˆ
φ
∫ ∫
1
ˆ
+
d
3
ˆ
ˆ
ˆ
x d
3
y v(|x − y|)φ
† (x)φ
† (y)φ(y)φ(x) (17.25)
2
ˆ
ˆ
where φ
† (x) creates a boson of mass m at position x. This H describes
identical bosons interacting via a potential v, which is assumed to be weak
(see, for example, Schiff 1968 section 55, or Parry 1973 chapter 1). We note
3 It is worth pausing to reflect on the idea that ordering is associated with symmetry
breaking.
/
17. Spontaneously Broken Global Symmetry
feature that for low q (long wavelength) their frequency ω tends to zero with
q (actually ω ∝ q
2 ). In this respect, therefore, they behave like massless particles when quantized – and this is another feature we should expect when a
symmetry is spontaneously broken.
The ferromagnet gives us one more useful insight. We have been assuming
that one particular ground state (e.g. the one with S z = N/2) has been somehow ‘chosen’. But what does the choosing? The answer to this is clear enough
ˆ
in the (perfectly r ealistic) case in which the Hamiltonian H S is supplemented
ˆ
by a term −gμB i S iz , representing the effect of an applied field B directed
along the z-axis. This term will indeed ensure that the ground state is unique,
and has S z = N/2. Consider now the two limits B → 0 and N → ∞, both
at finite temperature. When B → 0 at finite N , the N + 1 different S z eigenstates become degenerate, and we have an ensemble in which each enters with
an equal weight; there is therefore no loss of symmetry, even as N → ∞ (but
only after B → 0). On the other hand, if N → ∞ at finite B = 0, the single
state with S z = N/2 will be selected out as the unique ground state and this
asymmetric situation will persist even in the limit B → 0. In a (classical)
mean field theory approximation we suppose that an ‘internal field’ is ‘spontaneously generated’, which is aligned with the external B and survives even
as B → 0, thus ‘spontaneously’ breaking the symmetry.
The ferromagnet therefore provides an easily pictured system exhibiting
many of the features associated with spontaneous symmetry breaking; most
importantly, it strongly suggests that what is really characteristic about the
phenonenon is that it entails ‘spontaneous ordering’.
3 Generally such ordering
occurs below some characteristic ‘critical temperature’, T C . The field which
develops a non-zero equilibrium value below T C is called an ‘order parameter’. This concept forms the basis of Landau’s theory of second-order phase
transitions (see for example chapter XIV of Landau and Lifshitz 1980).
We now turn to an example much more closely analogous to the particle
physics applications: the superfluid.
17.3.2 The Bogoliubov superfluid
Consider the non-relativistic Hamiltonian (in the Schro ¨dinger picture)
∫
1
ˆ
H =
d
3
x
†
m
∇ ˆ
φ
2
· ∇ ˆ
φ
∫ ∫
1
ˆ
+
d
3
ˆ
ˆ
ˆ
x d
3
y v(|x − y|)φ
† (x)φ
† (y)φ(y)φ(x) (17.25)
2
ˆ
ˆ
where φ
† (x) creates a boson of mass m at position x. This H describes
identical bosons interacting via a potential v, which is assumed to be weak
(see, for example, Schiff 1968 section 55, or Parry 1973 chapter 1). We note
3 It is worth pausing to reflect on the idea that ordering is associated with symmetry
breaking.
/