Geometrical and Anderson transitions in harmonic chains with constrained long-range couplings

Low-dimensional systems with long-range couplings usually present phase transitions which are absent in the short-ranged counterpart model. In this work, we show that a harmonic chain with long-range couplings restricted by a cost function proportional to the chain length N exhibits two distinct phase transitions. In the present model, two sites at a distance r>1 are connected by a spring with probability 1/r(α) with the constraint that the total length of the non-nearest-neighbor couplings is limited to λN, where λ is a cost parameter. A geometrical phase transition is found at α=1.5 between a phase with a finite number of long-range couplings and a phase on which the number of long-range couplings is proportional to the system size. Further, the normal vibrational modes of this chain display a phase transition from delocalized to localized modes at a smaller value of α. Maximum effective disorder is reached at α=2 for which the frequency of the lowest vibrational mode exhibits a pronounced peak.