Quantum Rotation of the Néel Vector and Discrete Spin Waves
The ground state of a ring of magnetic moments (spins) with antiferromagnetic next-neighbor interactions can be displayed, in a classical picture, as a series of arrows pointing up and downwards:
Ground-state spin configuration of an antiferromagnet.
To obtain this representation, the spin ring was "cut" and then unfolded into
a linear chain (with periodic boundary conditions to reinforce the properties
of a ring).
Because of the antiferromagnetic interactions, the spins are collinear in the
ground state. Then the spin configuration can be represented by one single
vector of unit length, which for example might be chosen to be always
parallel to the leftmost spin in the above picture. This unit vector tracing the
coherent motion of the individual spins is called the Néel vector. A
Néel vector can be also defined quantum mechanically, but it becomes most
useful if the individual spins are at least that large that their dynamics is
well grasped by a semiclassical description. In a pure classical picture, as
used above (and in the following) the quantum mechanical zero-point motion is
neglected. But this is not too bad if the spins behave semiclassically, and
we can use our classical thinking.
Having an idea of the ground state, one can discuss the possible excitations.
One obvious excitation would be to "flip" one of the spins, or more precisely,
to change the magnetic quantum number M by one unit (for an up-spin e.g. from
M = s to M = s-1). For a spin with spin quantum number larger than s = 1/2,
this actually corresponds more to a canting of the spin than to a complete
flip. The energy required for such a "flip" is on the order
of twice the exchange energy as we have to break up two of the antiferromagnetic
bonds. However, importantly, such spin-flip states are NOT eigenstates of the
Hamiltonian (as one can choose any one of the N spins of the ring for the flip,
there are N such states). That is, the matrix elements
<spin n flipped|H|spin n+1 flipped> are not zero and the flip excitation
can "hop" along the ring from one site to the next. Because of the non-zero
hopping integrals of the localized spin-flip excitations, they have to be
combined to form delocalized, wave-like excitations characterized by a
wave vector k. These are the celebrated spin waves. The quantum theory for the
antiferromagnetic spin waves was given already more than 50 years ago in a
seminal paper of P. W. Anderson. The characteristic feature of the spin waves
is that, in a classical picture, the endpoints of the spin arrows rotate
around their up und down orientations, but such that the relative phases,
determined by the wave vector k, remain constant in time. This results in a
wave-like behavior, as it is shown here:
Spin-wave excitation in an antiferromagnet.
Energy wise, the spin waves form an energy band with excitation energies
e(q) = 2 s sin( 2pi/N q), where q is the shift quantum number (which is proportional
to the wave vector k) with values q = 0, 1, ..., N-1.
Apparently, the situation is very similar to what is happening in solid state
physics in the buildup of a crystal from individual atoms. Here, the localized
excitations are the electronic excitations of individual atoms. However, the more
the atoms approach each other, the more the atomic wave functions overlap (or more
precisely, the more the matrix elements <atom1|H|atom2>
increase), and the initially degenerate states spread out to form an electronic
band with the Bloch waves as resulting wave functions.
However, the spin waves are not the only possible excitations. In fact, there is a
second kind of excitation which is even lower in energy: the spins being parallel
in the ground-state can change their orientation in space coherently. That is, the
Néel vector may rotate. The required energy for that is lower than the
excitation of a spin wave since it is not necessary to break up any of the
antiferromagnetic bonds here. In our classical picture, the motion is such that the
spins always stay in parallel. This excitation mode may be picturized as follows:
Coherent motion of the ground-state spin configuration.
The long, dark green arrow to the right is supposed to represent the Néel
vector reflecting the coherent motion of the antiferromagnetic spin configuration,
though it is not of unit length here.
Importantly, this rotation of the Néel vector does not show up in extended
antiferromagnets. These systems undergo a phase transition to an ordered state, the
Néel state, for which the orientation of the sub-lattice magnetizations (and
wherby of the Néel vector) are fixed in space and time (here, for simplicity,
I have disregarded the issue of the dimensionality of the lattice for the moment).
That is, using more familiar wording, the system develops a spontaneous sublattice
magnetization or a symmetry-broken ground state, respectively. In order to change the
orientation of the sublattice magnetization (Néel vector), one has to apply
e.g. external magnetic fields to exert torques on the magnetization (Néel
vector) to force them to rotate.
It is interesting to analyze how it comes to this symmetry broken ground state. The
point is, that any finite Heisenberg antiferromagnet is invariant with respect to
rotations of the spins, i.e. of the magnetization ("finite" refers here to finite
number of spin centers of the Heisenberg ring). In the ordered state, which appears
only for the infinite system, however, this symmetry of the Hamiltonian is not reflected
anymore by the ground state. Following an argument given by P. W. Anderson, this appears
because the rotations which could restore the rotational symmetry become slower and
slower with increasing system size, and eventually become slower than any
experimental time scale - and whereby unobservable.
However, if the system is not infinite, then rotations of the sublattice magnetizations
or Néel vector, respectively, (recalling that the Néel vector is by
definition supposed to be collinear to the sublattice magnetizations) become observable.
And since the Néel vector actually represents quantum mechanical objects (the spins),
this rotation is quantized. In order to observe this quantized rotation, the system should
fulfill two requirements:
The ring should not be too large. For a finite, but still large system, the quantization
implies such small relative energy gaps, that the system is very well described in a
semiclassical picture. On the other hand, the system should not be too small, because
then the whole concept of a Néel vector becomes less apparent. In short, the system
should be of the "right" size.
The actual sample at hand should be well characterized as an antiferromagnetic Heisenberg
ring. This implies on the one hand that all the Heisenberg rings in the sample are equivalent
(monodisperse), and on the other hand that all the other magnetic effects like single-ion
anisotropies are much smaller than the Heisenberg interaction.
These conditions are not easily fulfilled simultaneously in reality, and only very few
systems are known to date, all of them being molecular wheels (as only these guarantee mono
dispersity). And one of them happened to be the Cr8 wheel...