200
17. Spontaneously Broken Global Symmetry
for some λ. However, in case (a) the charge annihilates a ground state, and
so all of them are really identical. In case (b), on the other hand, we cannot
ˆ
write (17.18) – since Q|0› does not exist – and we do have the possibility
of many degenerate ground states. In simple models one can verify that
these alternative ground states are all orthogonal to each other, in the infinite
volume limit – or perhaps more physically, the limit in which the number
of degrees of freedom becomes infinite. And each member of every ‘tower’
of excited states, built on these alternative ground states, is also orthogonal
to all the members of other towers. But any single tower must constitute a
complete space of states. It follows that states in different towers belong to
different complete spaces of states, that is to different – and inequivalent –
‘worlds’, each one built on one of the possible orthogonal ground states.
At first sight, a familiar example of these ideas seems to be that of a ferromagnet, below its Curie temperature T C . Consider an ‘ideal Heisenberg
ferromagnet’ with N atoms each of spin 1/2, described by a Hamiltonian of
ˆ ˆ
Heisenberg exchange form H S = −J S i · S j , where i and j label the atomic
sites. This Hamiltonian is invariant under spatial rotations, since it only
depends on the dot product of the spin operators. Suc h rotations are imˆ
plemented by unitary operators
S ·
ˆ
ˆ
exp(i
α) where S = i S i , and spins at
different sites are assumed to commute. As usual with angular momentum
in quantum mechanics, the eigenstates of H S are labelled by the eigen values
ˆ
ˆ
of total squared spin, and of one component of spin, say of S z =
i S iz .
The quantum mechanical ground state of H S is an eigenstate with total spin
quantum number S = N/2, and this state is (2 · N/2 + 1) = (N + 1)− fold
degenerate, allowing for all the possible eigenvalues (N/2, N/2 − 1, . . . − N/2)
ˆ
of S z for this value of S. We are free to choose any one of these degenerate
states as ‘the’ ground state, say the state with eigenvalue S z = N/2.
It is clear that the ground state is not invariant under the spin-rotation
symmetry of H S , which would require the eigenvalues S = S z = 0. Furthermore, this ground state is degenerate. So two important features of what
we have so far learned to expect of a spontaneously broken symmetry are
present – namely, ‘the ground state is not invariant under the symmetry of
the Hamiltonian’, and ‘the ground state is degenerate’. However, it has to
be emphasized that this ferromagnetic ground state does, in fact, respect the
symmetry of H S , in the sense that it belongs to an irreducible representation
of the symmetry group: the unusual feature is that it is not the ‘trivial’ (singlet) representation, as would be the case for an invariant ground state. The
spontaneous symmetry breaking which is the true model for particle physics
is that in which a many body ground state is not an eigenstate (trivial or
otherwise) of the symmetry operators of the Hamiltonian: rather it is a superposition of such eigenstates. We shall explore this for the superfluid and
the superconductor in due course.
Nevertheless, there are some useful insights to be gained from the ferromagnet. First, consider two ground states differing by a spin rotation. In the
first, the spins are all aligned along the 3-axis, say, and in the second along
17. Spontaneously Broken Global Symmetry
for some λ. However, in case (a) the charge annihilates a ground state, and
so all of them are really identical. In case (b), on the other hand, we cannot
ˆ
write (17.18) – since Q|0› does not exist – and we do have the possibility
of many degenerate ground states. In simple models one can verify that
these alternative ground states are all orthogonal to each other, in the infinite
volume limit – or perhaps more physically, the limit in which the number
of degrees of freedom becomes infinite. And each member of every ‘tower’
of excited states, built on these alternative ground states, is also orthogonal
to all the members of other towers. But any single tower must constitute a
complete space of states. It follows that states in different towers belong to
different complete spaces of states, that is to different – and inequivalent –
‘worlds’, each one built on one of the possible orthogonal ground states.
At first sight, a familiar example of these ideas seems to be that of a ferromagnet, below its Curie temperature T C . Consider an ‘ideal Heisenberg
ferromagnet’ with N atoms each of spin 1/2, described by a Hamiltonian of
ˆ ˆ
Heisenberg exchange form H S = −J S i · S j , where i and j label the atomic
sites. This Hamiltonian is invariant under spatial rotations, since it only
depends on the dot product of the spin operators. Suc h rotations are imˆ
plemented by unitary operators
S ·
ˆ
ˆ
exp(i
α) where S = i S i , and spins at
different sites are assumed to commute. As usual with angular momentum
in quantum mechanics, the eigenstates of H S are labelled by the eigen values
ˆ
ˆ
of total squared spin, and of one component of spin, say of S z =
i S iz .
The quantum mechanical ground state of H S is an eigenstate with total spin
quantum number S = N/2, and this state is (2 · N/2 + 1) = (N + 1)− fold
degenerate, allowing for all the possible eigenvalues (N/2, N/2 − 1, . . . − N/2)
ˆ
of S z for this value of S. We are free to choose any one of these degenerate
states as ‘the’ ground state, say the state with eigenvalue S z = N/2.
It is clear that the ground state is not invariant under the spin-rotation
symmetry of H S , which would require the eigenvalues S = S z = 0. Furthermore, this ground state is degenerate. So two important features of what
we have so far learned to expect of a spontaneously broken symmetry are
present – namely, ‘the ground state is not invariant under the symmetry of
the Hamiltonian’, and ‘the ground state is degenerate’. However, it has to
be emphasized that this ferromagnetic ground state does, in fact, respect the
symmetry of H S , in the sense that it belongs to an irreducible representation
of the symmetry group: the unusual feature is that it is not the ‘trivial’ (singlet) representation, as would be the case for an invariant ground state. The
spontaneous symmetry breaking which is the true model for particle physics
is that in which a many body ground state is not an eigenstate (trivial or
otherwise) of the symmetry operators of the Hamiltonian: rather it is a superposition of such eigenstates. We shall explore this for the superfluid and
the superconductor in due course.
Nevertheless, there are some useful insights to be gained from the ferromagnet. First, consider two ground states differing by a spin rotation. In the
first, the spins are all aligned along the 3-axis, say, and in the second along