重费米子材料是凝聚态物理强关联电子领域最受关注的材料之一。这一类材料上呈现出许多异常丰富的物理现象，如价健涨落、非常规超导和反常量子临界现象。对于这些现象的解释仍然是现今凝聚态理论学家的重要任务。周期性安德森模型是同时包含电荷和自旋自由度的、能够描述重费米子材料的最简单的模型。解析上缺乏严格解，数值上各种数值算法都有各自的瓶颈和缺陷。更多关于周期性安德森模型的研究工作仍然是必要的。本文采用了最新发展的数值方法-密度矩阵嵌入理论来研究二维正方晶格上的周期性安德森模型。 密度矩阵嵌入理论是2012年Garnet Chan小组提出的，可以同时应用到晶格模型上和量子化学计算中。在晶格模型上的计算，通过把晶格模型映射为一个杂质模型，对于杂质模型的严格求解来得到晶格上的物理量。由于杂质模型的大小仅仅是被选取的晶格模型上杂质格点数目的两倍，所需的计算量和计算资源极大地减小了，因而可以计算达到热力学极限尺寸大小的体系。与此同时计算精度也能够达到令人满意的程度。 我们首先研究了标准的周期性安德森模型。我们系统地研究了偏离半满时的基态相图，通过局域磁矩和自旋自旋关联函数的变化我们发现存在两个反铁磁相和一个顺磁近藤单态相。其中两个不同的反铁磁相之间是一级相变，是由费米面重构引起的Lifshitz相变。随着填充数接近半满，受电子-空穴对称性的约束，两个反铁磁相的区别消失，一级相变也不复存在。这个行为跟水的气液相变临界点的行为是一致的，填充数半满是一个量子临界点。我们还进一步分析了自旋-自旋关联函数，以及单重态和三重态上的投影算符。这些结果表明反铁磁相中同样也有近藤单态的形成，即反铁磁相与近藤单态相共存。 多轨道简并在重费米子材料上是一个普遍现象，我们还研究了一个格点上有两个简并$c$轨道和一个$f$轨道的周期性安德森模型。跟两个轨道的安德森模型相比，三个轨道的模型没有近藤能隙，在半满时是金属。我们计算了三个轨道的周期性安德森模型在半满时的相图，发现在半满时也存在Lifshitz相变。当两个$c$轨道和$f$轨道之间的杂化强度不等时，我们发现存在一个标度变换，这个标度变换可以使得杂化强度不同的情形变换到杂化强度相等时。不同轨道上的电子数、局域磁矩以及自旋-自旋关联函数都可以通过这个变换实现“数据碰撞“。

Heavy Fermion materials are one of the most notable class of materials in the field of strongly correlated electrons. They display many complex phenomena, such as valence fluctuation, unconventional superconductivity, abnormal quantum critical phenomena. The explanation of those phenomena is still one of the most important task for condensed matter theorists. Periodic Anderson Model (PAM) is the simplest model which contains both the charge and spin degree of freedom, and describes Heavy Fermion materials. Lack of exact solution from the analytical part, and each numerical methods has its own shortcomings. More work on PAM is still necessary. In this thesis we study the PAM on a two dimensional square lattice via the recently developed numerical method-Density Matrix Embedding Theory (DMET). DMET was proposed by Garnet Chan's group in 2012, and it could be applied to both lattice model and quantum chemistry calculation. For lattice model calculations it is by mapping the original lattice model to an impurity model, and then solve the impurity model accurately. Since the size of the impurity model is only twice of the impurity sites, the requirement of computational cost and computational resources is decreased intensively. Thus the thermodynamic limit could be reached. At the same time the computational accuracy is also satisfied. First we study the ordinary PAM. We systematically study the ground state phase diagram away from half filling. From the local magnetic moment and spin-spin correlation functions we found two anti-ferromagnetic phases and one para-magnetic Kondo singlet phase. The phase transition between the two AF phases are first order. It is Lishitz transition caused by Fermi surface reconstruction. As the filling approaches half filling, the difference between the two AF phases disappears, and the first order phase transition doesn't exist. This behavior is the same as the water liquid-vapor critical point, and half filling is a quantum critical point. We also analysis the spin-spin correlation functions and the projections to the spin singlet state and spin triplet states. Those results indicate the Kondo singlet is formed in the AF phase, which suggets AF phase and PKS phase are coexisted. The degeneracy of orbitals is common in heavy fermion material, thus we also study the 3 orbital PAM, in which there are two $c$ orbitals and one $f$ orbital on each site. Compared with the 2 orbital PAM, Kondo gap doesn't appear in the 3 orbtial PAM, and it's metal at half filling. We study the ground state phase diagram at half filling, and discovered Lifshitz transition appears even at half filling. When the hybridization strength of the two $c$ orbital and the $f$ orbital are different, there exists a scaling transformation, by applying which physical quantities are the same as when the hybridization is the same. Those quantities include number of electrons on different orbitals, local magnetic moments and the spin-spin correlation function.