這個論文探討凝態與極冷物理在低維度系統的多體現象。在光晶格的使用和實驗技術的推進影響下，晶格的幾何結構，相互作用和原子/分子的受局限動力學的交叉影響，引起許多吸引人的量子現象。進一方面，現在固態物理學中出現了許多新穎的二維材料。因為能帶接觸跟自旋偶和交互作用的共同影響，這些二維材料的能帶結構展現了不尋常的拓樸特性。無論是從學術上，或是潛在的運用上，我們從理論的角度切入，去探討了解相互作用，晶格的扭曲或是自旋跟軌道的糾纏所帶來的困難之處，並去追尋低維度系統的多樣性。這篇論文主要分成下面幾個章節:

第一章我們從不同的角度或領域方向簡介低維度物理。

第二章我們研究具有非等向性躍遷的二維波色哈伯模型(Bose-Hubbard model)。我們研究了非等向性對超流態的旋性模量(helicity modulus)以及普通流體到超流體(normal-to-superfluid)的Berezinskii-Kosterlitz-Thouless(BKT)相變溫度。我們使用兩種不同的方法: 一種是大尺度的量子蒙地卡羅的數值方法，另一種是自洽簡諧近似(self-consistent harmonic approximation,SCHA)。對於SCHA法，我們考慮了兩種極端情形:一個是二維各向同性還有一個是趨近一維極限的非等向性。我們發現SCHA提供了超流態非常合理的描述。特別的是，SCHA非常正確的描述一維極限的非等向性。在這個情形下，BKT溫度趨近於零(進入量子相變) 而且量子擾動也顯得特別重要。

第三章我們研究了在光晶格中兩分量波色子的p軌域波色愛因斯坦凝結。我們發展了新的虛時演化法來數值計算Gross-Pitaevskii方程式。此方法可以去除比 p能帶還要低的量子態，此外，此方法還可以運用到更高的軌域。我們的研究驗證了同種分量間的作用力喜歡複數的波色愛因斯坦凝結。此類波色愛因斯坦凝結具有交錯的軌道流量。更有趣的是，在不同種分量間的作用力比較弱的時候，複數的波色愛因斯坦凝結有兩種分類，一種破壞時間反演對稱性，另一種沒有。然而，當不同種分量間的作用力變強，這兩種複數波色愛因斯坦凝結會經歷一個量子相變，並變成一個實數，並擁有時間反演的波色愛因斯坦凝結。此時，此種波色愛因斯坦凝結具有交錯的自旋密度結構。我們探討了這種相變的起源，還有在實驗中可能的量測。

在第四章我們提出一個實驗手法，如何在一個多層系統中，有效地產生統一長度的偶極鏈氣體。如此實驗手法得到的偶極鏈可以形成一個偶極鏈晶體，而且其系統溫度很容易經由初始的晶格位能或是實驗過程中的外部電場強度來控制。當鏈的密度增加，我們可以進一步觀察到一個從偶極鏈晶體相解離為二維層結構晶體的二階量子相變。當量子漲落比經典能量重要時，壓縮係數會發散並決定這個二階量子相變的相邊界。我們討論這種偶極鏈晶體及其量子相變的實驗現象。

石墨烯的弱自旋軌道耦合可以經由吸附原子沉積大幅提高（像是Weeks等人所著作的物理評論文章 Phys. Rev. X 1， 021001 （2011） ） 。然而，吸附原子的動力學也會誘發聲子和電子自旋之間的耦合。在第五章中，我們用群論和緊束縛模型，系統地研究了在均勻吸附原子的單層石墨烯上，低能量的聲子和電子自旋如何耦合。我們的結果為這個系統的未來研究，像是自旋輸運和超導，奠定基礎。在受到這些聲子和電子自旋耦合影響下，為了量化電子自旋動力學，我們計算電子和空穴的自旋反轉率。我們展示自旋反轉率對於準粒子能量和系統溫度有很強的依賴性。

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.

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