# Tutorial

## 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...