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研究生: 高銘佐
Ming-Tso Kao
論文名稱: 空間各向異性與無序性之 (3+1)維量子海森堡模型的蒙地卡羅研究
Monte Carlo studies of (3+1)- dimensional quantum Heisenberg models with spatial anisotropy and disorder
指導教授: 江府峻
Jiang, Fu-Jiun
學位類別: 博士
Doctor
系所名稱: 物理學系
Department of Physics
論文出版年: 2014
畢業學年度: 103
語文別: 中文
論文頁數: 50
中文關鍵詞: 量子海森堡模型空間各項異性無序性蒙地卡羅模擬
英文關鍵詞: Quantum Heisenberg model, spatial anisotropy, disorder, Monte Carlo simulations
論文種類: 學術論文
相關次數: 點閱:217下載:25
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  • 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.

    口試委員會審定書 iii 誌謝 v Acknowledgements vii 摘要 ix Abstract xi 1 Introduction導論......................................1 2 Motivation 研究動機...................................7 3 Methods 研究方法......................................11 3.1 蒙地卡羅方法 (Monte Carlo Method)....................11 3.1.1 細致平衡與遍歷性 (Detailed balance and Ergodicity)..12 3.1.2 Metropolis 演算法.................................14 3.1.3 隨機級數展開法SSE..................................14 3.1.4 Measurement in SSE...............................23 3.2 誤差分析 Error analysis.............................23 3.2.1 數據分級 binning..................................24 3.2.2 Jackknife 分析................................... 24 3.2.3 Bootstrap 分析................................... 25 4 Numerical results數值結果.............................27 4.1 3d ladder空間各向異性之海森堡模型......................27 4.1.1 Néel 溫度 TN 的計算................................28 4.1.2 交錯磁化強度密度 Ms 的計算...........................30 4.1.3 自旋波速 c的計算...................................30 4.1.4 理論預測與蒙地卡羅的比較.............................32 4.2 disorder無序性......................................35 4.2.1 不同(D,P) 所對應之 Ms 和 s 的計算..................38 4.2.2 不同(D,P) 所對應之 TN 的計算 ......................39 4.2.3 TN 和 MS 之線性關係的驗證...........................41 5 Discussion and Conclusions 討論與結論..................45 Bibliography...........................................47

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