本文中，我们主要研究在磁场影响下，二维方晶格玻色子的基态量子相，以及在非厄米系统中玻色子生成的独特量子相。光晶格中超冷原子系统为模拟和探索凝聚态物理中的量子多体问题提供了纯净可靠的研究平台。此外，光晶格显示出高度的稳定性和更多的可调参数，有助于我们探索丰富的量子相。基于玻色-哈伯德模型，我们采用动态Gutzwiller平均场理论，通过分析所有格点的超流(superfluid，简称：SF)序参数和粒子密度分布来定量描述基态的量子相。

在第一章中，我们介绍了玻色-爱因斯坦凝聚(BEC)的研究背景、主要特点、理论框架和实验手段。此外，光晶格中引入人工规范场，可以实现不同规范场下磁通结构。我们还对超固相作出详细介绍。最后，我们给出非厄米系统的研究背景。在第二章中，我们研究在均匀人工磁通下，二维方晶格双组份玻色子的基态量子相特性。结果表明：当组份间相互作用较弱时(组份间相互作用小于位点相互作用)，出现特殊的量子相，如锯齿状密度波超固相(SuperSolid，简称：SS 相)、晶格状密度波SS相和超对流相(CounterFlow SuperFluid，简称CFSF)。此外，人工磁通的引入增大CFSF相的存在范围。当组份间相互作用较大时(组份间相互作用大于位点相互作用)，出现相分离的SS相(Phase Separated Supersolid，也称为P-SS相)。在第三章中，我们研究在交错磁通下，二维方晶格玻色子基态的量子相特性。结果表明：当交错磁通较大时，基态形成“创可贴”型SS相，即中间序参量的值较高，而左右(上下)的序参量略低，对应正-反涡旋的超级流分布。当交错磁通较小时，基态形成正-反涡旋“SF 凝聚相”，对应环形分布的超级流。此外，超级流密度的分布范围与跃迁强度的值成正比，而与交错磁通的值成反比。基于前两章的研究成果，在第四章中，我们引入非互易跃迁相因子的非厄米哈密顿量。基态量子相呈“SF岛”，即：MI海中表现出周期性分布的SF岛屿，且“SF岛”的数量与空间周期的平方成正比。在第五章中，我们进一步研究组份间相互作用对非厄米系统中双组份玻色子基态量子相的影响。当组份间相互作用较弱时，量子相包括“类涡状”的液体、“MI岛”、“SF岛”、以及CFSF 相。当组份间相互作用较强时，出现完全分离和不完全分离的“SF岛”。在最后一章中，我们总结研究内容，并对未来研究方向做出展望。我们考虑应用动态Gutzwiller平均场理论方法求解其他的等效晶格模型，如三角晶格中、六角晶格或者梯子模型中，探究是否会存在不同的量子相，以及是否可以尝试用精确对角化、或者蒙特卡罗数值计算进一步精确计算。

In this paper, we focus on the ground-state quantum phase of bosons in a two-dimensional square lattice under the influence of a magnetic field, as well as the unique ground-state quantum phase generated by bosons in non-Hermitian systems. The ultracold atomic system in optical lattices provides a clean and reliable research platform for simulating and exploring quantum many-body problems in condensed matter physics. In addition, the optical lattice exhibits high stability and a wide range of parameter adjustments, both of which contribute to our exploration of rich quantum phases. Based on the Bose-Hubbard model, we employ the dynamic Gutzwiller mean field theory to quantitatively describe the quantum phase of the ground state by analyzing the superfluid order parameters and particle density distributions of all lattice sites.

In chapter I, we review the research background, fundamental characteristics, theoretical frameworks, and experimental methods of Bose-Einstein condensation(BEC). Furthermore, we detail the realization of magnetic flux structures by incorporated different artificial gauge field in optical lattices, and we are described supersolid phase in detail. Finally, we also introduce the research background of non-Hermitian systems. In chapter II, we explore the quantum phase characteristics of bosons and boson mixtures in a two-dimensional square lattice under the influence of uniform artificial magnetic flux. In the case of weak inter-component interactions(component interaction is smaller than on site interaction), the quantum phases exhibit a jagged density-wave supersolid (SS) phase, a latticed-like density-wave SS phase, and a counterflow superfluid (CFSF). Furthermore, due to the presence of artificial magnetic flux, the coverage range of the CFSF phase is extended. On the other hand, for strong inter-component interactions(component interaction is bigger than on site interaction), the ground state of the Bose-Bose mixture system shows phase-separated supersolid (P-SS phase). In Chapter III, we investigate the quantum phase characteristics of the ground state of bosons in a two-dimensional square lattice under staggered magnetic flux conditions. The results show that when the staggered flux is large, the ground state forms a "band-aid" type SS phase, the value of the middle order parameter is higher, while the left and right (up and down) order parameter is slightly lower, aligning with the super-current's clockwise and anticlockwise vortex pairs. When the staggered flux is small, the ground state forms a positive and anti-vortex "SF condensed phase", corresponding to a circular super-current. Besides, the density of super-current is inversely proportional to the staggered magnetic flux, while it is positively related to the hopping strength. In chapter IV, based on the research results of the previous two chapters, we presents a non-Hermitian Hamiltonian incorporating a non-reciprocal hopping phase factor, resulting in zero magnetic flux. We introduce "SF islands", periodic arrangement of superfluid (SF) islands in the Mott insulator (MI) sea, with the number of "SF islands" is proportional to the square of the spatial period. In chapter V, we further explore the impact of inter-component interactions on the ground-state quantum phase of two-component bosons in non-Hermitian systems. In a regime of weak inter-component interactions regime, various fascinating quantum stages are noted, such as “vortex-like” liquid, “MI islands”, “SF islands”, and CFSF. In scenarios of strong inter-component interactions, we notice that the two-component bosons appear "SF islands" of complete and incomplete separation. In the last chapter, we provide a comprehensive summary of the research content and propose potential future research directions. We consider applying the dynamic Gutzwiller mean field theory method to other equivalent lattice models, such as triangular lattice, hexagonal lattice or ladder model, to investigate whether there are different quantum phases, and whether we can try to calculate them further with exact diagonalization or Monte Carlo numerical calculation.