# Is the phenomenon of geometrical frustration in condensed matter physics related to some kind of topological invariant?

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.

• Seems quite unlikely, although it is hard to disprove. As partial evidence: the only google result where "frustration" and "topological invariant" appear together is this post itself. Nov 12, 2021 at 1:22
• thoose really look like Wilson loops. so are wilson loops topological invariants? I think they are analogous to the quantized gauge charge of a compact gauge theory rather than topological invariants. Nov 16, 2021 at 1:41
• @physshyp Don’t know a lot about Wilson loops, but this Math Overflow question seems to suggest they’re related to topological invariants Nov 18, 2021 at 0:28
• @AccidentalFourierTransform Not so sure about that… I just found a pdf copy of the book “Geometrical Frustration” by Sadoc & Mosseri and it includes some discussions on, for instance, the “homotopic classification of defects”. Will try to read it carefully, but it would be nice to have an answer from a more knowledgeable person in the meantime. Nov 18, 2021 at 0:36
• Maybe it’s the sum of all the “frustration products” and not the frustration products themselves, that is related to some kind of topological invariant? Nov 18, 2021 at 0:41

I don't believe there is any known necessary topological invariant associated. Geometrical frustrated systems are simply systems where the "particles" get stuck in a configuration that is not good in terms of energy minimisation. This might seem a bit vague, but should make sense with an example.

The most obvious case is a array of spins with ferromagnetic interactions in 1st neighbors and anti-ferromagnetic with the 2nd neighbors. This leads to frustration and ultimatly to what is known as a spin glass (which yielded Geogio Parisi the Nobel in 2021.) I believe spin glasses to be the nicer example of geometrical frustration.

• Hi! Thank you for your answer. I did know about spin glasses and the basics of geometrical frustration –I once read Newman and Stein’s book ‘Spin Glasses and Complexity’– but I’m not sure this answers my question. Maybe it’s because, as @AccidentalFourierTransform said in the comments, it’s “hard to disprove” there is any topological invariant associated… but I still feel like there could be one, or that the local behaviour of geometrical frustration could globally relate to one, if that make sense. I’m thinking of something like a version of Gauss-Bonnet theorem. Maybe I’m just wrong though. Nov 24, 2021 at 11:52

You may find this recent Physical Review article interesting and related to your question here:

The Emergent Fine Structure Constant of Quantum Spin Ice Is Large

"Here, we show that the two greatly differ in their fine-structure constant α, which parametrizes how strongly matter couples to light: $${α_{QSI}}$$ is more than an order of magnitude greater than $$α_{QED}≈1/137$$."

They actually find in Spin Ice a fine structure constant of $${α_{QSI}}=1/10$$ !

There are some studies like these one here:

A topology for the electron

That correlates possible the fine structure constant of QED with geometry as a geometric proportionality constant and thus a geometric invariant in QED.

However we see in condensed matter geometric frustrated systems like Spin Ice this QED invariant, fine structure constant is heavily deformed in these strongly correlated electrons systems. Which may suggest a strongly deformed geometry of the dressed electron field.

So, the deformed fine structure constant in these systems maybe directly related to the geometric frustration phenomenon of these systems.

I don't know if this correlates with the winding numbers you are referring but I thought it was a good idea to mention this possible relation of the fine structure constant with the topological defects observed in these systems.

• Thank you, this looks interesting, will take a look! Nov 24, 2021 at 12:11
• @dahemar Also, the fact that topological defects on materials like geometrical frustration phenomenon in Spin Ice have experimentally been found to have a heavily altered fine structure constant value, strongly infers that the fine structure constant is in origin a geometric proportionality constant of charged particles like the electron. Nov 27, 2021 at 17:31
• @dahemar Notice, the fine structure constant found to be 1/10 for spin ice, will remain constant (invariant) for this material. Nov 27, 2021 at 17:37