199
17.3. Spontaneously broken symmetry in condensed matter physics
Returning to condensed matter systems, we introduce the BCS ground state
for a superconductor, in a way which builds on the Bogoliubov model of a
superfluid. We are then prepared for the application, in chapter 18, to spontaneous chiral symmetry breaking (question (i) above), following Nambu’s
profound analogy with one aspect of superconductivity. In chapter 19 we
shall see how a different aspect of superconductivity provides a model for the
answer to question (ii) above.
17.3 Spontaneously broken symmetry in condensed
matter physics
17.3.1 The ferromagnet
We have seen that everything depends on the properties of the vacuum state.
An essential aid to understanding hidden symmetry in quantum field theory
is provided by Nambu’s (1960) remarkable insight that the vacuum state of a
quantum field theory is analogous to the ground state of an interacting manybody system. It is the state of lowest energy – the equilibrium state, given
the kinetic and potential energies as specified in the Hamiltonian. Now the
ground state of a complicated system (for example, one involving interacting
fields) may well have unsuspected properties – which may, indeed, be very
hard to predict from the Hamiltonian. But we can postulate (even if we
cannot yet prove) properties of the quantum field theory vacuum |0› which
are analogous to those of the ground states of many physically interesting
many-body systems – such as superfluids and superconductors, to name two
with which we shall be principally concerned.
Now it is generally the case, in quantum mechanics, that the ground state
of any system described by a Hamiltonian is non-degenerate. Sometimes we
may meet systems in which apparently more than one state has the same
lowest energy eigenvalue. Yet in fact none of these states will be the true
ground state: tunnelling will take place between the various degenerate states,
and the true ground state will turn out to be a unique linear superposition
of them. This is, in fact, the only possibility for systems of finite spatial
extent, though in practice a state which is not the true ground state may
have an extremely long lifetime. However, in the case of fields (extending
presumably throughout all space), the Fabri–Picasso theorem shows that there
is an alternative possibility, which is often described as involving a ‘degenerate
ground state’ – a term we shall now elucidate. In case (a) of the theorem, the
ground state is unique. For, suppose that several ground states |0, a›, |0, b›, . . .
existed, with the symmetry unitarily implemented. Then one ground state will
be related to another by
iλ ˆ
|0, a› = e
Q
|0, b›
(17.18)
17.3. Spontaneously broken symmetry in condensed matter physics
Returning to condensed matter systems, we introduce the BCS ground state
for a superconductor, in a way which builds on the Bogoliubov model of a
superfluid. We are then prepared for the application, in chapter 18, to spontaneous chiral symmetry breaking (question (i) above), following Nambu’s
profound analogy with one aspect of superconductivity. In chapter 19 we
shall see how a different aspect of superconductivity provides a model for the
answer to question (ii) above.
17.3 Spontaneously broken symmetry in condensed
matter physics
17.3.1 The ferromagnet
We have seen that everything depends on the properties of the vacuum state.
An essential aid to understanding hidden symmetry in quantum field theory
is provided by Nambu’s (1960) remarkable insight that the vacuum state of a
quantum field theory is analogous to the ground state of an interacting manybody system. It is the state of lowest energy – the equilibrium state, given
the kinetic and potential energies as specified in the Hamiltonian. Now the
ground state of a complicated system (for example, one involving interacting
fields) may well have unsuspected properties – which may, indeed, be very
hard to predict from the Hamiltonian. But we can postulate (even if we
cannot yet prove) properties of the quantum field theory vacuum |0› which
are analogous to those of the ground states of many physically interesting
many-body systems – such as superfluids and superconductors, to name two
with which we shall be principally concerned.
Now it is generally the case, in quantum mechanics, that the ground state
of any system described by a Hamiltonian is non-degenerate. Sometimes we
may meet systems in which apparently more than one state has the same
lowest energy eigenvalue. Yet in fact none of these states will be the true
ground state: tunnelling will take place between the various degenerate states,
and the true ground state will turn out to be a unique linear superposition
of them. This is, in fact, the only possibility for systems of finite spatial
extent, though in practice a state which is not the true ground state may
have an extremely long lifetime. However, in the case of fields (extending
presumably throughout all space), the Fabri–Picasso theorem shows that there
is an alternative possibility, which is often described as involving a ‘degenerate
ground state’ – a term we shall now elucidate. In case (a) of the theorem, the
ground state is unique. For, suppose that several ground states |0, a›, |0, b›, . . .
existed, with the symmetry unitarily implemented. Then one ground state will
be related to another by
iλ ˆ
|0, a› = e
Q
|0, b›
(17.18)