在过去二十年的理论研究表明，材料不能简单的分为绝缘体和导体，一些材料具有内部绝缘、边界导电的性质，这与占据能带的拓扑结构有关。因此，本文对于具有这种性质的一维和二维拓扑绝缘体材料进行建模和计算。

对于一维晶格，我们通过一个具体的系统:Su-Schrieffer-Heeger(SSH)模型来求解薛定谔方程，并定义一维晶格的拓扑不变量——卷绕数。随后我们利用了一个具体的公式来实现对卷绕数的计算，并通过对于SSH模型卷绕数的不同以及边界态的性质，实现了对于一维晶格的拓扑态和平凡态分类。

对于二维晶格，本文主要考察二能级系统。我们在二维晶格模型中引入了类似一维晶格的拓扑不变量——陈数，并实现了对陈数的计算。随后我们对于一个算例（Qi-Wu-Zhang模型）进行数值模拟，通过原始定义我们可以计算陈数随模型的参量变化的情况。而针对这种方法计算慢、有奇点和没有普适性的问题，我们又设计了一种仅通过特征向量计算陈数的算法，使得计算量减少，最后我们还实现了二维晶格多能级系统的陈数的计算。

In the past two decades, theoretical research has shown that materials cannot be simply divided into insulators and conductors, and some materials have the properties of internal insulation and boundary conduction, which is related to the topology of the occupying band. Therefore, in this paper, we model and calculate the one - and two-dimensional topological insulator materials with this property.

For one-dimensional lattices, we solve the Schrodinger equation through a concrete system: the Su-Schrieffer-Heeger(SSH) model, and define the topological invariant of the one-dimensional lattice - the winding number. Then we use a specific formula to calculate the winding number, and through the difference of the winding number of the SSH model and the properties of the boundary state, we realize the classification of the topological state and the trivial state of the one-dimensional lattice.

For two-dimensional lattices, this paper mainly investigates two-level systems. We introduce a topological invariant similar to a one-dimensional lattice, the term number, into the two-dimensional lattice model, and realize the calculation of the term number. Then we perform a numerical simulation for an example (Qi-Wu-Zhang model), and through the original definition we can calculate the change of the number of terms with the parameters of the model. In order to solve the problem of slow computation, singularity and no universality, we design an algorithm to calculate the term number only by eigenvector, which reduces the computation amount. Finally, we also realize the term number calculation of two-dimensional lattice for multi-level system.