随着实验技术的进展，近几十年来，人们又开始关注凝聚态系统的零温量子相变的研究。本文主要是从数值和解析两个方面研究了二维 Wen-plaqutte 模型加入横向外场后的量子相变。本文第一章，主要介绍了量子相变研究的进展和近几十年来凝聚态物理的进展。第二章介绍了量子相变的一些理论，并着重介绍了量子伊辛模型（即横场伊辛模型）的量子相变理论。在第三章，我们主要通过数值方法研究了二维 Wen-plaqutte 模型加入横向外场后的量子相变。在这一章，首先介绍了 二维 Wen-plaqutte 模型提出的理论背景及这个模型的一些性质，这是一个严格可解模型，它的基态由投影对称群（PSG）描述。接下来介绍了我们所用的数值方法：严格对角化方法及 lanczos 方法，并较详细的说明了这两个方法在我们所研究的问题中的应用。利用数值方法，我们验证了二维 Wen-plaqutte 模型的基态简并度、基态能量、以及第一激发态与基态的能隙；研究了不同格点模型的基态能量随外场参数的变化情况。通过分析第一激发态与基态的能隙与外场参数J的曲线，基态的磁化强度随外场参数$J$变化的曲线，以及二维 Wen-plaqutte 模型中的F项的基态平均值关于外场参数J变化的曲线，我们得到结论：量子相变发生在J/|g| = 1.0点。发生量子相变前后，模型的基态从拓扑序态变为自旋极化态。在本文第四章，我们采用了一个对偶变换，将二维 Wen-plaqutte 模型对偶到了一维横场伊辛模型，我们通过计算在量子相变临界点一维横场伊辛模型中的关联函数而得到二维 Wen-plaqutte 模型中“F”的的关联函数。

As the technology of experiment is developing , the past decade has been a substantial rejuvenation of interest in the study of quantum phase transitions. We study the quantum phase transition of Wen plaqutte model in a transverse field in two dimensions by both numerical method and analytic method .In the first chapter of this thesis, we give an introduction to the quantum phase transition and show its applications on condensed matter physics. In the second chapter, the theoretical development of quantum phase transitions are introduced. In particular, the quantum phase transition in the quantum Ising model (the transverse Ising model) are discussed in detail. In the third chapter, we study the quantum phase transition of the Wen-plaqutte model in a transverse field by numerical method. In this chapter, the physical properties for Wen plaqutte model in two dimensions is reviewed firstly. This model is an exact soluble model, whose ground state is characterized by projective symmetry group(PSG). We used the exact diagonalization method and the lanczos method to learn the nature of the topological order for its ground states, including the topological degeneracy, the energy of the ground state and the energy gap between the first excited state and ground state. By analysing the energy gap between the first excited state and the ground state, the magnetization of the ground state( varied as J), and the average value of F term, we draw a conclusion that the quantum phase transition occurs at the point J/|g| = 1.0. As the quantum phase transition occurs by raising the strength of the transverse field, the ground state changes from a topological ordered state to a spin polarized state. In the fourth chapter, the two dimensional Wen-plaqutte model in a transverse field is mapped to the transverse Ising model by a dual transformation. We obtain the correlation function of F in Wen plaqutte model at quantum critical point by calculating the correlation function of transverse Ising model.