微分流形上量子可积系统的精确量子化和热力学方法

近年来，微分流形上的低维量子多体系统成为凝聚态理论研究的热点问题之一，相关结果在其它领域例如非平衡态统计、高能、弦论和数学物理等都有重要应用。本项目拟研究此类系统的精确量子化和热力学。具体研究思想是，替代传统的Bethe根表示，我们将提出系统转移矩阵本征值的T-W方法，利用本征值的零根和W函数的零根来刻画能谱和本征态。在此基础上，利用零根Bethe ansatz方程，得到体系精确的态密度，进而研究标度律和热力学，计算可观测物理量。具体研究内容包括：（1）莫比乌斯带上的精确量子化、马约拉纳零模、扭曲关联态和共形荷等；（2）扭曲微分流形上的统计模型和平衡态过程；（3）多维Torus上的高秩和超对称量子可积系统；（4）扭曲微分流形上的精确自由能、有限尺寸效应、临界指数和热力学。通过本项目的执行，期望能得到一些重要的物理机理、物理图像和普适代数结构，加深人们对量子多体系统的理解，促进本领域的发展。

Recently, the study of low-dimensional quantum many-body systems in the differential manifolds becomes one hot issue in condensed matter theory. Meanwhile, the related results have important applications in other research fields such as non-equilibrium statistics, high energy, string theory and mathematical physics. In this project, we will study the exact quantization and thermodynamics of such systems. The specific research idea is as follows. Instead of Bethe roots representation, we will propose the T-W method for the eigenvalues of transfer matrix of the systems. Then, we will use the zero roots of the eigenvalue and the zero roots of W function to describe the energy spectrum and the corresponding eigenstates. Based on these facts and using the Bethe ansatz equations satisfied by the zero roots, we will calculate the analytical expression of the density of state. Then we can study the scaling behavior and thermodynamics. We will compute the exact free energy and calculate some observable physical quantities such as the specific heat and the magnetic susceptibility. The specific research contents are shown as follows. (1) The exact quantization, Mayorana zero modes, twisted correlated states and conformal charges of the quantum integrable systems in the Mobius band; (2) Statistical models and equilibrium states processes in the twisted differential manifolds; (3) Quantum integrable systems with high-rank or super-symmetries in the high-dimensional torus; (4) The finite size effects, exact free energy and thermodynamics of the system in the twisted differential manifolds. From these studies, we want to obtain some important physical mechanisms, physical pictures and universal algebraic structures. We also want to deepen people's understanding of quantum many-body systems and promote the development of this field.