Geometrical Frustration - Mathematical Definition

Mathematical Definition

The mathematical definition is simple (and analogous to the so-called Wilson loop in Quantum chromodynamics): One considers for example expressions ("total energies" or "Hamiltonians") of the form

where G is the graph considered, whereas the quantities are the so-called „exchange energies“ between nearest-neighbours, which (in the energy units considered) assume the values (mathematically, this is a signed graph), while the are inner products of scalar or vectorial spins or pseudo-spins. If the graph G has quadratic or triangular faces P, the so-called "plaquette variables", "loop-products" of the following kind, appear:

resp.

which are also called "frustration products“. One has to perform a sum over these products, summed over all plaquettes. The result for a single plaquette is either +1 or -1. In the last-mentioned case the plaquette is "geometrically frustrated".

It can be shown that the result has a simple gauge invariance: it does not change – nor do other measurable quantities, e.g. the "total energy" – even if locally the exchange integrals and the spins are simultaneously modified as follows:

Here the numbers und

are arbitrary signs, i.e. = +1 or = −1, so that the modified structure may look totally random.