t-J 模型相變之研究與高磁場下量子點特性之探討

Abstract

本論文首先研究二維 t-J 模型的反鐵磁態之相圖，接著探討強磁場下量子點的基態性質。在t-J 模型中，我們利用奴隸玻色子(slave boson)的技巧來處理每個晶格點只能填一個電子的問題。在這方法下，以前的研究提出了一些可能的序參量來描述基態的性質，這些序參量包括了超導序(或共振價鍵序)與磁通量序等。這些序參量雖然能定性的解釋高溫超導的一些實驗結果，但確不能解釋在零電洞濃度附近高溫超導體的反鐵磁性。 我們除了考慮以上兩種序參量外，又考慮了反鐵磁序參量，並在平均場的近似下研究這些序參量隨溫度與電洞濃度的改變情形。我們發現這些序參量之間的關係是共存而非競爭的，而反鐵磁態的相變溫度對電洞濃度關係也與實驗相符合。 在量子點的問題中，我們首先注意到了電子在高磁場下的量子點中和它在量子霍爾效應中所處的情況之類似性，因而提出了 Skyrmion 當做量子點基態的試探波函數。 Skyrmion 的特性可由一拓樸不變量 : 繞旋數來描述，我們試了各種繞旋數的Skyrmion ，並以最低能量者當成基態，進而得知基態如何隨一些實驗參數，如磁場、侷限位能而改變。這些試探波函數的能量與一些理論結果比教起來是相當好的，而我們得到的化學位能對磁場的關係圖也與實驗吻合，並能解釋最近實驗所觀測到的新相變。

In this dissertation, we will study the antiferromagnetic phase diagram of t-J model and then study the ground state properties of a quantum dot in high magnetic field. We use the technique of slave boson to cope with the no double occupancy constraint in t-J model. There are some researchers pointed out that some order parameters such as superconductivity order (or resonance valence bond order) and flux phase order that can characterize the properties of the ground state of t-J model. Although these order parameters can explain some experimental results of high temperature superconductivity qualitatively, they fail in the explanation of the antiferromagnetic properties around zero doping. In addition to the two order parameters, we take the antiferromagnetic order parameter into consideration and study the relationships between theses order parameters and temperature or doping concentration. We found that these order parameters behave cooperatively rather than competitively and the relationship of Neel temperature vs. doping concentration is consistent with experiments. In the study of quantum dots, we note that the electrons in the environments quantum dots under high magnetic field are similar to the electrons in the environments of quantum Hall effect. Therefore we propose the skyrmion as the trial wave finction as the ground state of quantum dot in high magnetic field. A skyrmion is characterized by a topological conserved quantitity, winding number, and therefore we try skyrmions with various winding numbers and take the one with the smallest winding number as the ground state. The properties of skyrmions that changed with magnetic field and confining potential are obtained. The energies of our trial wave functions are lower than many previous works. The phase diagram of chemical potential vs. magnetic field is consistent with experimental results and it can explain the new found ground states transitions observed in recent experiment.