Edit (attempt to clarify my question a little bit):
I’m not thinking geometrical frustration should be necessarily associated to a topological invariant in a direct way, but maybe local geometrical frustration behaviour correlates to one globally, in a way akin to how curvature and the Euler characteristic of a manifold are related by Gauss-Bonnet-Chern theorem? Not sure what kind of topological invariant or what topological space would these be.
@physshyp pointed out in the comments that the mathematical description of this phenomenon is very similar to Wilson loops in gauge theories. Well, I unfortunately don’t know anything about Wilson loops, but this MathOverflow question cites “Wilson loops averages” as a known topological invariant. Moreover, the definition of these already kind of reminds me of the statement of Gauss-Bonnet-Chern. I guess this might be a coincidence though! But I’d like to know how this translates to geometrical frustration, in case it does.
Original question:
(Hi, I asked this on Math Exchange but didn’t have any replies so I was wondering if maybe it belongs here).
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 $I_{k_ν,k_μ}$ 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 $S_{k_ν}·S_{k_μ}$ 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.
