I have studied Ising model and in general interactions inside the Hamiltonian of the form $ \sum_{ijk}k_{ijk}S_{i}S_{j}S_{k}$ (where k is the coupling constant) are disregarded. What I don't understand is how such interactions which are NOT pairs exist. For example in classical mechanics I have personally never seen a force which instead of acting in pairs acts in groups of 3 particles. How can I visualise an interaction like this?
Thanks!
I know of three-body or many-body interactions that are consequence of an effective description of the dynamics of one system, given some conditions. I'll sketch some argument about how it appears.
We can have a clue that these interactions should exist, at least effectively. Consider the time Evolution operator in interaction picture, of a system governed by some Hamiltonian $H = H_0 + V(t)$. We have
$$ U(t) = \mathbb I - \frac i\hbar \int_0^t dt_1 V_I (t_1) + \left( \frac i\hbar \right)^2 \int_0^t dt_1 \int_0^{t_1} dt_2 V_I(t_1)V_I(t_2) + \mathcal O(3). $$
Now, if the potential $V$ is some two-body potential like $V = g(t)\sum_{jk} S_jS_k$, all the terms of second or high order in the expression above will have terms of three or more interactions. For simplicity, if $G(t)$ is the integral of $g(t)$ and $G(t)\approx \sqrt t$,
$$ U(t) \approx \mathbb I - \frac i\hbar \sqrt t \sum_{jk}S^I_jS^I_k + \left( \frac i\hbar\right)^2 t \sum_{jklm} S^I_jS^kS^I_lS^I_m + \mathcal O(3). $$
In some cases, the second term r.h.s. could be neglected, e.g. if we are interested in interaction between specific subspaces of the space of states where it acts trivially. The term proportional to $t$ acts like an effective Hamiltonian
$$ U(t) \approx \mathbb I - \frac i\hbar t \mathcal H_{eff} + \dots $$
And in this case, it would be a four-body interaction.
I didn't worked it carefully, but you could find concrete examples elsewhere. Despite that the original Hamiltonian of the system only includes two-body interactions, a system could have many-body interactions effectively.
