# Parity of a 1d Ising model, and with higher order terms

I don't know if this should be asked here or in a math stack exchange, but I'll try here first.

Consider the classical 1d Ising model with periodic boundary condition: $$$$H_2 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i \sigma_{i+1}$$$$ where $$\sigma_i=\pm1$$, and we refer to the collective $$(\sigma_1,\sigma_2,\ldots,\sigma_n)$$ as $$\vec{\sigma}$$.

Now, if there exists a configuration $$\vec\sigma$$ such that $$H_2(\vec\sigma)=0$$, that actually means $$n$$ is divisible by $$4$$. Mathematically, $$$$\exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_2(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 4.$$$$ This is because all two-body terms multiplies to $$1$$, i.e. $$G:=\prod(\sigma_i\sigma_{i+1})=1$$, and therefore there are even number (say, $$2k$$) of terms that are $$\sigma_i\sigma_{i+1}=-1$$. The LHS condition says there should be the same amount of -1 terms as the +1 terms. This means the total number of terms $$n$$ must be double of the number of -1 terms, i.e. $$n=4k$$.

This was an innocent exercise that resulted from the PBC, but maybe we can also view it as an "extension" from the following more trivial fact. Consider the 1-body version of the Hamiltonian: $$$$H_1 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i .$$$$ Now, we have a more trivial statement that looks similar to the previous one. $$$$\exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_1(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 2.$$$$

If we observe the trend here, it is tempting to guess that the following is true. Define $$$$H_4 (\vec{\sigma}) = \sum_{i=1}^n \sigma_i\sigma_{i+1}\sigma_{i+2}\sigma_{i+3} .$$$$ Then, the following holds? $$$$\exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_4(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 8.$$$$ However, this turns out to be false! The correct statement actually is $$$$\exists \vec\sigma ~~~\mathrm{s.t.}~~~ H_4(\vec\sigma)=0 ~~~~~~ \Leftrightarrow ~~~~~~ n\equiv 0 \mod 4 ~~~ \wedge ~~~ n\neq 4 ,$$$$ which seems rather random.

All of this seems to be potentially connected to deeper ideas like gauge symmetry etc. because for example, the first mod $$4$$ result heavily relies on the PBC, which is about the topology of the system. Also, in papers that talk about connections between stat-mech models and gauge redundancies, Kramers-Wannier dualities, the $$G=1$$ seems to be crucial (e.g. in https://arxiv.org/pdf/2310.16032.pdf appendix B-I they discuss that).

My question is this:

1. Do people talk about Hamiltonians like $$H_4$$? It looks like the 1d version of the $$\mathbb{Z}_2$$ lattice gauge theory to me. But is that even a thing? Is there a "better" way to generalize $$H_1$$ and $$H_2$$ so that the result is more consistent?
2. Perhaps in the math field this is a thing with a name like boolean analysis. Does anyone know any reference for those things?

Thanks!

• Mathematical physicists occasionally consider Ising-type Hamiltonians of the form $\sum_{A\subset \{1,\dots,n\}} J_A \sigma_A$, where $(J_A)_{A\subset\{1,\dots,n\}}$ is a collection of real numbers and $\sigma_A = \prod_{i\in A} \sigma_i$. This class of Hamiltonians certainly includes those you are interested in. This class is natural when studying, for instance, the GKS inequality. See, for example, the book Group Analysis of Classical Lattice Systems by Gruber et al for much more about such systems. Commented Feb 29 at 9:11
• Thanks @YvanVelenik for your reference! Commented Mar 1 at 0:12