Abstract
This thesis reports on the study of many-body phenomena in low-dimensional systems in condensed matter and ultra-cold physics. With use of optical lattice potentials and enabling experimental techniques, many intriguing quantum phenomena have arisen from interplay between lattice geometry and interactions with the confined dynamics of the atoms/molecules. On the other hand, a wide family of novel two-dimensional materials in solid state physics are available nowadays, exhibiting non-trivial topological properties of their band structure, which are caused by a combination of band-close and spin-orbital coupling. From the virtue of purely academic purpose or the view of potential applications, our works mainly pursue variations in low-dimensional systems in an effort to theoretically understand challenges involved with interactions, lattice distortions, or even the entanglement of the spin and orbital degrees of freedom. This PhD thesis will be divided into following chapters: In Chapter 1, we very briefly introduce the low dimensional systems in various aspects and fields. In Chapter 2, we study the two-dimensional Bose-Hubbard model with anisotropic hopping. Focusing on the effects of anisotropy on the superfluid properties such like the helicity modulus and the normal-to-superfluid (Berezinskii- Kosterlitz-Thouless, BKT) transition temperature, two different approaches are compared: Large-scale Quantum Monte Carlo simulations and the self-consistent harmonic approximation (SCHA). For the latter, two different formulations are considered, one applying near the isotropic limit and the other applying in the extremely anisotropic limit. Thus we find that the SCHA provides a reasonable description of superfluid properties of this system provided the appropriate type of formulation is employed. The accuracy of the SCHA in the extremely anisotropic limit, where the BKT transition temperature is tuned to zero (i.e. into a Quantum critical point) and therefore quantum fluctuations play a dominant role, is particularly striking. In Chapter 3, we investigate the unconventional Bose-Einstein condensations (BECs) of two-species mixture with the p-wave symmetry in the second band of a bipartite optical lattice. A new modified imaginary-time propagation method is developed to numerically solve the Gross-Pitaevskii (GP) equation by truncating states in the lowest bands, and can be applicable to even higher orbital bands. Different from single-species case, the two-species boson mixture exhibits two non-equivalent complex BECs: One breaks timereversal symmetry but one does not, in the dominant intra-species interaction regime. When the inter-species interaction is turned stronger, both states undergo a quantum phase transition at the SU(2) invariant point toward a real-valued checkerboard state with a staggered spin density structure. We also discuss the lattice asymmetry, strong interaction effect and experimental implication. In Chapter 4, we propose an experimental scheme to effectively assemble chains of dipolar gases with an uniform length in a multi-layer system. The obtained dipolar chains can form a chain crystal with the system temperature easily controlled by the initial lattice potential and the external field strength during process. When the density of chains increases, we further observe a second order quantum phase transition for the chain crystal to be dissociated toward layers of 2D crystal, where the quantum fluctuation dominates the classical energy and the compressibility diverges at the phase boundary. Experimental implication of such dipolar chain crystal and its quantum phase transition is also discussed. The naturally weak spin-orbit coupling in Graphene can be largely enhanced by adatom deposition (e.g. Weeks et al. Phys. Rev. X 1, 021001 (2011)). However, the dynamics of the adatoms also induces a coupling between phonons and the electron spin. In Chapter 5, using group theory and a tight-binding model, we systematically investigate the coupling between the low-energy in-plane phonons and the electron spin in single-layer graphene uniformly decorated with heavy adatoms. Our results provide the foundation for future investigations of spin transport and superconductivity in this system. In order to quantify the effect of the coupling to the lattice on the electronic spin dynamics, we compute the spin-flip rate of electrons and holes. We show that the latter exhibits a strong dependence on the quasi-particle energy and system temperature.