论文标题
Lieb-Schultz-Mattis定理：拓扑观点
The Lieb-Schultz-Mattis Theorem: A Topological Point of View
论文作者
论文摘要
我们回顾了Lieb-Schultz-Mattis定理及其变体，这些变体是不可定理的，表明具有某些条件的量子多体系统不能具有本地唯一的间隙基态状态。我们将自己限制在一维量子自旋系统上，并讨论了具有U（1）对称性的模型的广义Lieb-Schultz-Mattis定理，以及具有离散对称性的模型的扩展Lieb-Schultz-Mattis定理。我们还讨论了相同的参数对系统在无限缸上的影响，无论是在周期性的边界条件和螺旋边界条件下。 对于具有U（1）对称性的模型，我们在这里提出了基于Twist Operator的Lieb，Schultz和Mattis的原始证明的重排版本。正如标题所暗示的那样，我们采取了现代的拓扑观点，并通过使用拓扑指数（与填充因子相吻合）来证明广义的Lieb-Schultz-Mattis定理。通过拓扑指数，我们的意思是一个索引，该指数表征了局部唯一的基地面状态，并且在基态的连续（平滑）修饰下是不变的。 对于具有离散对称性的模型，我们根据在受对称保护的拓扑阶段中提出的拓扑指数中描述了最通用的证据的基本思想。我们从背景材料开始，例如对称组的投射表示分类。 我们还回顾了这样的概念，即我们称之为无限晶格上量子自旋系统的本地唯一间隙基态并呈现基本定理。从物理学家的角度来看，这个概念是自然而有用的。 我们试图使本文可读并且几乎独立。我们仅假设有关量子自旋系统的基本知识。
We review the Lieb-Schultz-Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb-Schultz-Mattis theorem for models with U(1) symmetry and the extended Lieb-Schultz-Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests we take a modern topological point of view and prove the generalized Lieb-Schultz-Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists' point of view. We have tried to make the present article readable and almost self-contained. We only assume basic knowledge about quantum spin systems.
