包含U(1)对称性破缺或者粒子数不守恒的系统在物理上是广泛存在的，传统方法在在处理这类系统时无能为力。本项目拟利用近年来新发展的Off-diagonal Bethe Ansatz方法在处理U(1)对称破缺这类可积模型上的优势，研究凝聚态、量子场论中的可积模型，主要关注非平庸边界条件导致(准)粒子数不守恒的系统。研究这类模型的精确解，是从系统转移矩阵出发计算算子层次的恒等式，进而得到哈密顿量本征值的T-Q函数关系，得到能谱，分析其本征态的性质。应用和推广Off-diagonal Bethe Ansatz方法，将现有的方法推广至高秩及无穷维的情形。发展相关的非对角情形的热力学Bethe Ansatz方法，用以研究这些模型的热力学性质，计算热力学配分函数、关联函数，并探讨其热力学极限下的性质。

Systems without U(1) symmetry or particle number conserving are widespread in Physics, the traditional method is powerless in dealing with such systems. This project aims to make use of the superiority of newly developed Off-diagonal Bethe Ansatz method in dealing with U(1) symmetry broken systems, to study the integrable models in condensed state and quantum field theory. Mainly concern about the systems without (quasi) particle numbers conservation caused by nontrival boundaries. Firstly, start with the transfer matrix, calculate the operator identities and the T-Q functions of the Hamiltonian eigenvalues, derive their energy spectrum and then analyse properties of the eigenstates under this circumstance. Applying and popularizing Off-diagonal Bethe Ansatz method, extend the existing method to the case of high rank and infinite dimension. We develop related non diagonal thermodynamic Bethe Ansatz methods to study thermodynamic properties of these models, calculate thermodynamic partition functions and correlation functions, and discuss their properties under thermodynamic limit.