Collective properties of quantum matter: from Hawking radiation analogues to quantum Hall effect in graphene

Abstract

The work is divided in three parts. We devote the first part to the study of analog Hawking radiation in Bose-Einstein condensates. We study numerically the birth of a sonic black hole in an outcoupled Bose-Einstein condensate after relaxing the confinement provided by an optical lattice. We also study possible signatures of spontaneous Hawking radiation. We propose that the violation of CS inequalities is a smoking gun of the presence of the Hawking effect. We compare this criterion with the presence of entaglement, finding that both are equivalent under usual assumptions. Finally, we study a different gravitational analogue: the so-called black-hole laser. The most interesting result is the appearance of a regime of continuous and periodic emission of solitons, providing the most strong analogue with optical lasers. In the second part, we analyze the effect of the introduction of a short Bragg pulse in a thermal cloud. We show that the induced periodic density pattern decays to the equilibrium profile. However, instead of the usual collisional relaxation, the mechanism responsible for the decay is the thermal disorder of the particles, with a characteristic time that only depends on the temperature. We find a very good agreement with actual experimental data. In the last part, we switch to a very different system: the $\nu=0$ quantum Hall state of bilayer graphene. After re-deriving the corresponding mean-field phase diagram, we compute the collective modes within the zero Landau level. Among the most remarkable results, we have found that at the boundary between the FLP and the F phases a gapless mode appears resulting from an accidental symmetry that can be regarded as a remanent of a broken $SO(5)$ symmetry. On the other hand, the CAF and PLP phases can present dynamical instabilities. We straightforwardly extend the previous results to monolayer graphene.