在本文中，我们借助量子蒙特卡罗随机级数展开('Stochastic Series Expansion'，简称'SSE')方法研究了Kagome晶格上的Kitaev-Heisenberg模型的相图。讨论了零温情形Mott绝缘体相-超流体相-Mott绝缘体相的量子相变，并确定了临界磁场h_{c1}=1.011(3)，h_{c2}=1.952(2)，以及量子相变的临界指数\eta与\nu，但是并没有发现量子自旋液体的存在。对于h_{c1}<h<h_{c2}时的超流体相，高温不稳定，随着温度升高，会发生Kosterliz-Thouless相变，超流体密度“跳跃”到0。本文中也确定了不同磁场强度下的临界温度T_{KT}，从而给出了整体相图。本文结构安排如下
       第一章
介绍了相关背景知识，并对期待发现的量子自旋液体进行了简单介绍。然后介绍了外加磁场下Kagome晶格中的Kitaev-Heisenberg模型，并介绍了计算中采取的自旋旋转操作。
       第二章
介绍了研究中使用的量子蒙特卡罗SSE方法，以及相关物理量在该算法中的具体计算形式。
       第三章
 介绍了Kagome晶格中Kitaev-Heisenberg模型的量子蒙特卡罗研究结果。并分别介绍了零温量子相变与有限温Kosterliz-Thouless相变结果，最后给出了外加磁场下Kagome晶格上Kitaev-Heisenberg模型的整体相图。

 

      In this thesis,we study the phase diagram of the Kitaev-Heisenberg model on the Kagome lattice using the quantum Monte Carlo 'Stochastic Series Expansion'method(SSE) .We discuss the transition of Mott insulator-super fluid-Mott insulator when the temperature is zero. We get the critical magnetic field h_{c1}=1.011(3)，h_{c2}=1.952(2) and calculate the critical exponents \eta and \nu. But we don't find quantum spin liquid in this model. For h_{c1}<h<h_{c2}, the super fluid will be destroyed when the temperature raise. There is a Kosterliz-Thouless transition and the super fluid density will "jump" to zero. We calculate the critical temperature T_{KT} under different external magnetic field and get the phase diagram of the Kitaev-Heisenberg model on the Kagome lattice.

      In chapter one, the background of our research work is introduced. We are looking forward to find quantum spin liquid in our model, so the related background is introduced. Then we calculate the Hamiltonian of the Kitaev-Heisenberg model on the Kagome lattice under the external magnetic field. To use SSE, we introduce the method of rotating spin.
      In chapter two, we introduce the quantum monte carlo method SSE used in our research and the measurement of the related quantities using SSE.
      In chapter three, we discuss the numerical results of the Kitaev-Heisenberg model. We respectively introduce the quantum transition when temperature is zero and the Kosterliz-Thouless  transition when temperature raise. Finally, we get the phase diagram of the Kitaev-Heisenberg model on Kagome lattice under the external magnetic field.