In discrete models like Ising we have Hamiltonians of the form
$$H(\sigma)=\frac{1}{N}\sum_{i=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$$
where $\sigma_{i}=\pm 1$ , $J_{ij}$ interaction coefficients and N is size of system.
So I was wondering if anyone has studied the following Hamiltonian
$$\sum_{i=1}^{N}J_{ij}e^{ix_{i}}e^{ix_{j}},$$
where $x_{i}\in [0,2\pi]$. So the phase space is $\mathbb{T}^{N}$. Then we can ask similar questions, like what is the Gibb's measure at $\mathbb{T}^{\infty}$ and so the distribution of the free energy $\frac{1}{N}ElogZ_{N}$.
Note that I am not taking a dot product like in the Heisenberg model.
Also, note the interesting picture of the Hamiltonian $\sum_{i=1}^{3}e^{ix_{i}}e^{ix_{j}}$. Eventually it fills the entire 3-disk, but I thought it was pretty cool seeing those cluster points.
Any references will be greatly appreciated.
