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# Geometrical frustration

Geometrical frustration is a phenomenon in condensed matter physics in which the geometrical properties of the atomic lattice forbid the existence of a unique ground state, resulting in a nonzero residual entropy. The most important consequence of this is that the entropy of the system does not go to zero at absolute zero. An example of a geometrically frustrated material is ordinary water ice, which has a residual entropy of around 3.40 J/K/mole.

Geometrical frustration is an important feature in magnetism, where it arises from the topographic arrangement of spins. One simple example is shown in Fig 1. Three magnetic ions reside on the corner of a triangle with antiferromagnetic interactions between them. Once the first two spins align anti-parallel, the third one is frustrated because its two orientations (up or down) give the same energy - the third spin cannot simultaneously minimise its interactions with both of the other two. Thus the ground state is twofold degenerate.

Geometrical frustration can also arise when four spins are arranged in a tetrahedron (Figure 2). If there is an antiferromagnetic interaction between spins (i.e. a spin wants to point in the opposite way to its neighbour), then it is not possible to arrange the spins so that all interactions between spins are antiparallel. In the diagram shown there are six interactions between neighbours, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and 3 and 4) are unfavourable. Thus, once again it is impossible to have all interactions favourable, and the system is frustrated.

Geometrical frustration is also possible if the spins are not colinear. If we consider a tetrahedron with a spin on each vertex pointing along the 'easy axis' (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net overall spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each and every spin, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours; that is, we want to maximise spins. The best situation possible is shown in Figure 4, with two spins pointing towards the centre and two pointing away. Effectively, all the arrows are pointing upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out (i.e. are antiferromagnetically aligned), as do forwards and backwards.

A consequence of geometrical frustration is that the lowest energy state can be achieved in a number of fashions. If we consider the system in figure 3, there is only one way to arrange the four arrows so that they all point inwards; the ground state in this unfrustrated system is singly degenerate. Considering figure 4, there are three different ways to arrange the fours spins in a two-in, two-out fashion, so this frustrated state has a three-fold degeneracy. This is exactly the situation found in spin ices, and is mathematically analogous to the arrange of hydrogen atoms in ice.