EMERGENT COLLECTIVE PHENOMENA IN QUANTUM MANY-BODY SYSTEMS

Motivated by a recent proposal by Lev and coworkers, in the first part of this thesis, we will perform a theoretical investigation of a new class of quantum simulators: the so called multimode disordered Dicke simulators. Our approach is mostly inspired from Statistical Mechanics: indeed we will merge together exact results obtained in the context of the Dicke model by Hepp and Lieb, with known results on disordered systems and neural networks. In this way we will be able to generalize the standard approach to the superradiant phase transition to the disordered case. As a byproduct of this analysis we will argue that this new class of quantum simulators (properly engineered) may be an alternative (or complementary) route toward quantum computation. Also the second part of the thesis has a ”quantum simulators motivation”. Recently Bloch’s group implemented an Ising quantum magnet with long-range antiferro- magnetic interactions, which exhibits a peculiar devil’s staircase phase diagram, predicted long ago by Bak and Bruinsma. This result, joined with recent theoretical investigations by Lesanowsky and coworkers suggested to us to reconsider these spin models in the context of the fractional quantum Hall effect (FQHE). In the second part of this thesis we will show that the quantum Hall Hamitonian projected on the lowest Landau level can be mapped, in the so called thin torus limit, on the lattice gas studied by Bak and Bruinsma. This observation will lead us to predict a devil’s staircase scenario for the Hall conductance as a function of the magnetic field. This work stimulated us to investigate the connection between Laughlin wave function and Tao-Thouless states, that we will explore in the last section of the second part.