本论文主要基于玻色-哈伯德模型及其扩展模型，研究磁性光格子中量子气的性质及有效模型。由于冷原子系统具有高度可控的特性，不同相互作用下的哈伯德模型能够在实验中得以实现。而理论上可以探讨各种效应对超流与莫特绝缘相的影响，从而研究新奇的物相与有趣的现象。基于此，我们将探究人造规范场作用下有效磁通对光晶格中边缘手征流的影响、特殊晶格模型中的能带拓扑特征以及莫特绝缘体中的磁性相结构。

在第一章中，我们首先介绍光晶格中的超冷玻色原子气，包括玻色-爱因斯坦凝聚理论背景、实验实现过程，并对光晶格系统的实现以及紧束缚晶格模型的理论进行阐述。其次我们以 Su–Schrieffer–Heeger 模型及其扩展模型为例介绍一维系统中的能带拓扑，以磁通梯子晶格为例介绍基态下的手征流特性，以自旋-轨道耦合效应为例介绍旋量玻色气体的低能物理。

第二章至第四章为主要研究内容，具体介绍如下：

（一）在双链晶格中，随着磁通的增加可能导致手征流发生反转，这一现象来自于相互作用引起的原胞合并。在无相互作用的双链晶格中，无法通过调节晶格参数观测到手征流的反转。我们将手征流的研究扩展到三链模型中，该模型相当于一个具有三个内部自由度的玻色子一维晶格。我们重点研究均匀磁通作用下三链晶格中无相互作用玻色子在基态下的手征流分布特征，通过控制引起各链之间二次塞曼位移的偏置场，探讨了晶格中手征流的反转现象。我们基于能带理论分析手征流的反转条件。同时我们揭示了与迈斯纳-涡旋相变有关或无关的三种流反转机制。而在弱相互作用条件下，手征流的反转几乎不受影响。

（二）我们研究了一个扩展的 Creutz 梯子模型，该系统的能带可能表现出非平庸拓扑属性。其拓扑相变可由拓扑不变量的突变和赝自旋的不同构型揭示。在沿梯子方向的线性势作用下，我们进一步研究了拓扑能带上的布洛赫振荡。在拓扑相变点，由于能带交叉，振荡周期加倍，波包在两个能带上交替演化。布洛赫振荡中赝自旋极化的大小可以为能带拓扑特征的动态识别提供依据。我们将 $y$ 和 $z$ 轴的局部赝自旋极化与可测量的链间流和密度差进行了联系，展示了布洛赫波包的微观运动和非绝热演化特征，研究了波包在两链间的微小振荡。此外，还探讨了在能带近交叉点 (avoided band crossing) 附近的 Landau-Zener 隧穿效应，并给出了隧穿概率的表达式。

（三）研究规范场存在下的莫特绝缘体，即强相互作用玻色-哈伯德模型的有效自旋模型。我们首先介绍一种基于绝热消除的简单方法来确定低能有效哈密顿量，并在一个具有有效磁通的双链模型中进行演示。我们进一步考虑一个扩展的玻色-哈伯德模型，该模型涵盖了自旋轨道耦合和有效磁通的情形。得到的 XYZ 模型包含 Dzyaloshinskii-Moriya 相互作用以及对称各向异性耦合项。在特定条件下，该模型可简化为各种具有丰富相图的自旋哈密顿量。我们研究了两种典型情况下的序参量和自旋构型，得到了基态相图。其中的数值模拟利用了张量网络算法（虚时演化分块抽取和变分矩阵积态方法）。

最后在第五章中，我们对本论文的研究内容和意义进行详细的总结，并对后续的研究思路和方向做了展望。

In this thesis, we mainly focus on the properties and effective models of lattice quantum gas in the presence of synthetic flux. Our study is based on Bose-Hubbard model and its extensions. Such models with varying interactions are readily realized in cold atom experiments. This is due to the high level of controllability of lattice gases. Theoretically, intriguing phases and phenomena appear when considering various effects in both superfluid and Mott insulating phases. Specifically, We investigate chiral currents in two-leg and three-leg ladders with synthetic flux. we further explore the magnetic phases of spinor Bosonic gases in the Mott regime. The band topology of a generalized Creutz ladder is also studied.

In chapter I, we first introduce the ultra-cold Bosonic gas in optical lattice, including the background of Bose-Einstein condensation, experimental realizations, and the basic features of optical lattice. The band theory and Hubbard model are also discussed. We introduce the extended Su-Schrieffer-Heeger model and present the concepts of band topology as well as chiral currents. We further make a brief review of the effect of spin-orbit coupling in optical lattice.

Chapter II  to Chapter IV are the main contents of our research, which are arranged as follows:

In the two-leg ladder lattice, the increase of the magnetic flux may lead to the inversion of the chiral current, which is due to the combination of cells caused by the interaction. However, the inversion of the chiral current cannot be observed by adjusting lattice parameters in a non-interacting two-leg ladder system. We extend the study of chiral current to a three-leg ladder model which is equivalent to a one-dimensional lattice of bosons with three internal degrees of freedom. For non-interacting bosons in a three-chain lattice with uniform magnetic flux, we focus our discussions on the ground state chiral current distribution. Then we discuss the inversion of chiral current by controlling the bias field that causes a quadratic Zeeman shift between the legs. We analyze the conditions for the inversion of chiral current based on band theory. At the same time, we reveal three current reversal mechanisms related or unrelated to the Meissner-vortex phase transition. The inversion of chiral currents is almost unaffected in the presence of a weak interaction.

We study an extended Creutz ladder system with distinct diagonal couplings. Such a model supports topological phase transition which is revealed by the change of topological invariant and pseudospin texture. With a linear force along the ladder, typical Bloch oscillations appears. At the topological phase transition point, the oscillation period is doubled due to band crossings and the wave packet evolves alternatively on the two bands. The topological feature can be captured by the magnitude of pseudospin polarizations. The local pseudospin polarizations along $y-$ and $z-$ axis are directly related to inter-leg current and density difference. Beyond the description of Bloch oscillation, we observe a small density oscillation which is due to the micromotion of the Bloch wave packet. In addition, we show non-adiabatic propagations at avoided band crossings in Bloch oscillation. The transition probability of the Landau-Zener tunneling is also obtained.

We study the effective spin models for Mott insulating Bose-Hubbard model in the presence of gauge fields. We first introduce a simple method based on adiabatic elimination to determine the low energy effective Hamiltonian. Demonstration of a two-component 1D model with synthetic magnetic field is shown. We further consider a generalized Bose-Hubbard model which covers the cases where spin-orbit coupling and synthetic flux are present. The resulting XYZ model contains Dzyaloshinskii-Moriya interactions and symmetric anisotropic couplings. Under certain conditions, this model reduces to various interesting magnetic Hamiltonians. Phase diagrams for two typical situations are investigated. Results are validated by numerical calculations using infinite Time-Evolution Block Decimation and variational Matrix-Product-State methods.

Finally, a summary of the results in this thesis and an outlook for the future research are included in chapter V.