Quantum entanglement in random and inhomogeneous spin chains

Résumé

Entanglement can be quantified in bipartite systems using the entanglement entropy which is, in many cases, bounded by an area law. Nonetheless, some critical 1D systems violate this area law and present logarithmic corrections which are parametrized by the central charge of the associated conformal field theory (CFT). Furthermore, other violations to the area law may appear in random systems and inhomogeneous systems. In the first part of the work we focus on the properties of entanglement in random hopping models, focusing on the similarities between the CFT predictions for the clean case and the strong disorder renormalization group (SDRG) predictions. We use a combination of methods: exact diagonalization, a transformation of the SDRG and a new tool based on the study of random permutations. All techniques coincide in providing a compelling image, based on a bond-picture. The second part focus on the engineering of an inhomogeneous fermionic system, in order to obtain a maximal growth of the block entanglement entropy in the ground state. We show that the deformation parameter allows us to interpolate between an uniform model and the strong disorder limit, which is a maximally entangled state called the rainbow state. This state can be seen as a thermo field state. The last part focus on time-evolution of some valence bond states after a global quench to a homogeneous Hamiltonian in 1D. We focused on the evolution of the entanglement in the rainbow and dimer states. We proposed a generalization of the quasiparticle picture, which we have called the ballistic picture.