Dealing with a zombie in EU law: the regulatory procedure with scrutiny

Annotation to Joined Cases T-261/13 and T-86/14 Netherlands v. Commission, EU:T:2015:67

A quantum topological phase transition at the microscopic level

We study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entanglement and the topological entropy of the system as a function of this coupling constant, and show that the topological entropy remains constant all the way up to the critical point, and jumps to zero beyond it. Despite the jump in the topological entropy, the transition is second order as detected via any local observable.Comment: (13 pages, 4 figures) v2: updated references and acknowledgments; v3: final update (references) after publicatio

Exact results on the quench dynamics of the entanglement entropy in the toric code

We study quantum quenches in the two-dimensional Kitaev toric code model and compute exactly the time-dependent entanglement entropy of the non-equilibrium wave-function evolving from a paramagnetic initial state with the toric code Hamiltonian. We find that the area law survives at all times. Adding disorder to the toric code couplings makes the entanglement entropy per unit boundary length saturate to disorder-independent values at long times and in the thermodynamic limit. There are order-one corrections to the area law from the corners in the subsystem boundary but the topological entropy remains zero at all times. We argue that breaking the integrability with a small magnetic field could change the area law to a volume scaling as expected of thermalized states but is not sufficient for forming topological entanglement due to the presence of an excess energy and a finite density of defects.Comment: 14 pages, 7 figures, published versio