Phase Transitions And Their Controllability In Quantum Many-body Systems

There have always been two topics in the research of condensed matter. The first one is the topic of energy band theory and perturbation theory, which are closely related to the Landau theory of Fermi liquid. This theoretical system is the cornerstone of the semiconductor theory, which has supported the current research of various kinds of electronic devices. The other one is the Landau theory of symmetry breaking and the renormalization group theory. This theoretical system provide us a frame to study the state of matter and its phase transitions between different states, and it is also the cornerstone of many fields of technology, such as the liquid-crystal displays, the magnetic record materiel, the alloy material as well as the polymers and so on. The two topics are usually complementary and overlapping sometimes. According to Landau’s symmetry-breaking theory, both the thermal phase transition induced by the thermal fluctuation and the quantum phase transition (QPT) induced by the quantum fluctuation would give rise to changes of the system’s order parameters as well as the symmetry. In Chapters2and3of the present thesis, we would discuss the thermal phase transitions and quantum phase transitions behaviors of some systems under this frame. In Chapter4, we would discuss another type of QPT without local order parameter and thus no symmetry breaking was realized. This new type of QPT is named as topological QPT for the topological behavior of its ground state. We would employ the geometric phase to discuss the topological critical behaviors in the Kitaev honeycomb model. Furthermore, we finally generalize the geometric phase, which has been successfully used to characterize the QPTs in general system, into the topological QPT system.In the last decade, due to the quantum many-body systems of ultracold atoms can be precisely controlled experimentally, and therefore seem to provide an ideal platform on which to study the properties of the quantum many-body systems, it has become a research hotspot to realize and even control these physical effects and theoretical models experimentally. In Chapters2and3, our work is mainly based on the famous Dicke model, whose phase transition from a normal phase to a superradiant phase has been observed experimentally in2010. Particularly, we in Chapter2discuss the ground state behaviors and the quantum criticality of Dicke model in the representation of spin coherent state. We in this Chapter obtained the ground-state energy and the occupation-number difference between the two levels of the atoms; as well as the quantum phase transition in a cavity optomechanics with a Bose-Einstein condensate based on an extended Dicke model is also been discussed. In Chapter3, we mainly study the thermal behaviors in an equal Rashba and Dresslhaus spin-orbit coupled Bose-Einstein condensate, which has been realized in an ultracold neutral atomic system. Properties of the specific heat and the entropy are obtained as well. Finally, we propose a scheme to apply this system into the field of quantum information. According to our theory, we can obtain a giant spin squeezing with factor over-30dB, and furthermore, to choose a proper phase factor of the prepared initial state can also enhance the squeezing factor. In Chapter4, we demonstrate that the ground state geometric phase obtained via the correlated rotation can be used to characterize the topological QPT in the Kitaev honeycomb model.