The two-body random spin ensemble and a new type of quantum phase transition
We study in this paper the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the 'entangling' part of the interaction. We then specialize to a spin chain with nearest-neighbour interactions and numerically find a new type of quantum-phase transition related to the strength of a random external field, i.e. the time-reversal-breaking one-body interaction term.
We study the Darboux integrability of the Moon–Rand polynomial differential system. Moreover we study the limit cycles of the perturbed Moon–Rand system bifurcating from the equilibrium point located at the origin, when it is perturbed inside the class of all quadratic polynomial differential systems in R3, and we prove that at first order in the perturbation parameter e the perturbed system can exhibit one limit cycle, and that at second order it can exhibit four limit cycles bifurcating from the origin. We provide explicit expressions of these limit cycles up to order O(e2).