Why do we go beyond two-body interaction? Actually, my question is why do we study many-body interactions. I have just started working in Fractional quantum Hall systems. There we have Coulomb interactions between electrons, which we know is a 2-body interaction. But then we go further and study >2 body interactions. If Coulomb interaction is 2-body, what is causing >2 body interactions?
 A: In condensed matter, n-body interactions are the rule more than the exception. How do they appear, taking into account that at a fundamental level the relevant interaction is the pair-wise Coulomb interaction? 
Simply because in many cases there is need of averaging over some of the the degrees of freedom. As a  prototype example, let's look how one can derive an inter-atomic interaction between atomic degrees of freedom, by "averaging" over the electronic degrees of freedom (basically the Born-Oppenheimer approximation).
No doubt that the complete form of the interaction Hamiltonian can be written as a sum of pair-wise interactions: nucleus-nucleus, nucleus-electron and electron-electron Coulomb interactions. However solving the resulting Schrödinger equation would be a very complex task. In many cases it is possible to justify reduced description where one splits the problem in two steps


*

*evaluate the electronic ground state $E_{GS}( \left\{ {\bf R_i} \right\}  )$ at fixed nuclei positions $\left\{ {\bf R_i} \right\}$;

*treating the nuclei as interacting via an interaction which is 
$$
U(\left\{ {\bf R_i} \right\} )= \sum_{i<j} \frac{Z_iZ_j}{\left| {\bf R_i-R_j}  \right|} +E_{GS}( \left\{ {\bf R_i} \right\}  )
$$
In general, there is no physically justified way to write $E_{GS}( \left\{ {\bf R_i} \right\}  )$ as a sum of pair-wise terms. On the contrary, there are direct experimental proofs that the total potential energy cannot be put in the form of a sum of pairwise terms. 


A simple signature comes from elasticity theory: in a cubic crystal, for symmetry reasons, there should be three independent elastic constants. However, if the interaction potential is pair-wise, there is a relation among the three constants (Cauchy relation) and one would remain with only two independent elastic constants. Experiments tell us that Cauchy relation is violated in the majority of the experimental systems. 
