We consider one dimensional systems of particles interacting with each other through long range interactions that are translation invariant. We seek quasi-periodic equilibrium states.
Standard arguments show that if there are continuous families of quasi-periodic ground states, the system can have large scale motion, if the family of ground states is discontinuous, the system is pinned down. The transition between the two cases is called breakdown of analyticity and has been widely studied.
We use recently developed fast and efficient algorithms to compute all the continuous families of ground states even close to the boundary of analyticity. We show that the boundary of analyticity can be computed by monitoring some appropriate norm of the computed solutions.
We implemented these algorithms on several models. We found that there are regions where the boundary is smooth and the breakdown satisfies scaling relations. In other regions, the scalings seem to be interrupted and restart again. We present a renormalization group explanation of these phenomena. This suggest that the renormalization group may have some complicated global behavior.
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