I have a bit of background (not at a very high level) in both math and physics and was recently reading about the phenomenon of geometrical frustration which appears in some materials where, according to Wikipedia, “atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favoring rather simple, but different structures) lead to quite complex structures.”
The fact that this phenomenon seems to be of a “global” rather than “local” nature made me wonder if it’s related to some kind of topological invariant associated to these lattices. However, the mathematical discussion included in the Wikipedia article didn’t mention that.
In particular, this part of the discussion reminded me of winding numbers:
The mathematical definition is simple (…). One considers for example expressions ("total energies" or "Hamiltonians") of the form
$$ \mathcal {H}=\sum _{G}-I_{k_{\nu },k_{\mu }}\,\,S_{k_{\nu }}\cdot S_{k_{\mu }} $$
where $G$ is the graph considered, whereas the quantities $I_{kν,kμ}$ are the so-called "exchange energies" between nearest-neighbours, which (in the energy units considered) assume the values ±1 (mathematically, this is a signed graph), while the $S_{kν}·S_{kμ}$ 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" {PW}, "loop-products" of the following kind, appear:
$$P_W=I_{1,2}\,I_{2,3}\,I_{3,4}\,I_{4,1}$$ and
$$P_W=I_{1,2}\,I_{2,3}\,I_{3,1}$$
respectively, 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".
Excuse-me if the question is a bit vague, but I don’t know how to formulate it more precisely.