拓扑量子态是当前凝聚态物理研究的前沿问题之一，在实验和理论上不断显现出内涵丰富的新奇性质。本项目主要研究低维关联多体系统中的边界自由度、聚合特性、拓扑表面态及其标度行为，所采用的方法主要是精确解，它可以为新物理概念和物理图像提供基准。研究内容是：一维关联多体系统如Hubbard模型和超对称t-J模型等在非平庸拓扑边界条件下的边界能、表面态和关联函数等。具体来讲：（1）利用聚合和相应的代数结构，寻求系统转移矩阵满足的封闭的算子恒等式，进而得到系统的能谱和非齐次Bethe ansatz方程；（2）研究体系的热力学极限，计算体系的边界能、表面态、标度率，建立相应的普适类；（3）利用非均匀参数，构造体系希尔伯特空间的完备基，利用已经求得的本征值，得到相应的本征态；（4）严格求解系统的形状因子和关联函数，计算关联长度，考虑相互作用对边界态的影响。通过对这些问题研究，希望能建立某些新的物理图像和机理。

The study of topological quantum states is one of the frontier topics in condensed matter physics. Many interesting phenomena and new physics have been obtained in this field, both in experiments and in theories. In this project, we will study the degree of freedom at boundary, fusion, topological surface states and their scaling behavior in the low-dimensional correlated many-body systems. The main method is the exact solution, which can provide benchmark for some new physical concepts and new physical pictures. The main research contents are the boundary energy, the surface state and the correlation function of some one-dimensional many-body systems, such as the Hubbard model and the generalized supersymmetric t-J model. The detaield research plan is as follows. (1) By using the fusion and the analysis of algebraic structure, we want to obtain some closed operator identities of the fused transfer matrices. Based on them, we can obtain the energy spectrum of the system and inhomogeneous Bethe nsatz equations; (2) We will study the properties of the systems in the thermodynamic limit, including the surface state, the scaling law and the universal class of the boundary; (3) By using the inhomogeneous parameters, we will construct the complete basis of the Hilbert space of the system. With the help of obtained eigenvalues, we will construct the eigenstates of the system; (4) We will study the form factor, correlation function, correlation length, and the effects induced by the interactions on the boundary states. Through the study of these problems, we hope to establish some new physical pictures and new physical mechanism.