本論文主要是使用蒙地卡羅方法 (Monte Carlo Method) 來模擬研究 (3+1) 維量子海森堡模型 (quantum Heisenberg model)。特別是我們探討了空間各向異性 (spatial anisotropy)與無序性 (disorder)對此模型特性之影響。
研究空間各向異性量子海森堡模型的動機是想要針對 dimerization 類別的海森堡模型，定量上去探討在量子臨界點附近 (quantum critical point) 新建立的普適關係 (universal relation)，即 $T_N/\sqrt{c^3}\propto\sf{ M_s}$ 。其中 $T_N$ 是 Néel temperature ，$c$ 是自旋波速 (spin wave velocity)及 $M_s$ 是交錯磁化密度 (staggered magnetization density)。
我們所作的模擬結果與 Sushkov \cite{Sushkov:2012:PRB} 藉由級數展開法 (series expansion) 所得到的結果是一致的。
另外對無序性的研究，我們計算三維鍵結無序 (bond disorder) 量子海森堡模型的 $\overline{T_N}$ 和 $\overline{M_s}$ ，方法是引進兩個參數，即隨機耦合強度 $D$ 和隨機機率 $P$ ，來描述反鐵磁交換耦合 (exchange couplings) $J_{ij}$ 的隨機性。$D$ 和 $P$ 的值皆在 $0$ 和 $1$ 之間，每個交換耦合強度為 $J_{ij}(1+D)$ 或 $J_{ij}(1-D)$ 的機率分別為 $P $ 及 $(1-P)$ 。 我們發現對這種無序性模型在靠近乾淨系統附近，用平均交換耦合強度 $\overline{J}$ 歸一化的 $\overline{T_N}$ (即 $\overline{T_N}/\overline{J}$) 和交錯磁化密度 $\overline{M_s}$ 之間也呈現一種線性關係。

In this thesis we simulate the three-dimensional quantum Heisenberg model using first principles Monte Carlo method. We focus on the effects of spatially anisotropy and random-exchange disorder. Our motivation is to investigate quantitatively the newly established universal relation $T_N/\sqrt{c^3}\propto\sf{ M_s}$ near the quantum critical point (QCP) associated with dimerization. Here, $T_N$, $c$ and $\sf{M_s}$ are the Néel temperature, the spin wave velocity and the staggered magnetization density, respectively. Our Monte Carlo results agree nicely with the corresponding results determined by the series expansion method. As for the random-exchange disorder, the randomness for the antiferromagnetic exchange couplings $J_{ij}$ (bond disorder) for any two nearest neighbour spin $\langle ij \rangle$ is introduced by two parameters $D$ and $P$. Specifically, given a set of $0<P<1$ and $0<D<1$, the probability that each antiferromagnetic coupling takes the value $J_{ij}(1+D)$ ($J_{ij}(1-D)$) is $P$ ($1-P$). Remarkably, in contrast to the scenario of the dimerized systems that the linear relation between $T_N$ and $M_s$ appears close to a quantum critical point at which the antiferromagnetism is destroyed, for the model considered here the Néel temperatures, when being normalized properly, scale linearly with the staggered magnetization density near the data associated with the clean system. Our study also confirms that in three dimensions the antiferromagnetism is robust against the employed bond disorder.

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