我们引入了分层对称算符和分层异常，作为量子晶格系统中本地对称算符和本地投影表示的推广。分层对称算符是一种可以分解为互不依赖子系统对称算符的对称算符；其分层异常定义为与这些子系统算符相关的异常的集合。我们发展了内部对称性的分层异常的细胞链复形形式化，证明在存在晶格对称性的情况下，它们会导致Lieb-Schultz-Mattis（LSM）约束。这包括LSM异常和SPT-LSM定理。我们将这一框架应用于调制G对称性，这些对称性的总对称群为G_tot = G ⋊ G_s，其中G_s是晶格对称群。值得注意的是，一个非零分层异常在G_s的基本域内（例如，一个单位元）并不总是意味着调制对称性的LSM异常。相反，存在LSM异常也取决于G_s如何作用于G。当G_s是晶格平移群时，我们找到了分层异常导致LSM异常的显式判据，并利用G_s不变的细胞链的同调群来分类LSM异常。我们通过具有分层异常的指数和偶极对称性的例子在(1+1)D和(2+1)D中进行了说明，并构建了一个受SPT-LSM定理约束的调制SPT稳定器码模型。
英文摘要：
We introduce stratified symmetry operators and stratified anomalies in quantum lattice systems as generalizations of onsite symmetry operators and onsite projective representations. A stratified symmetry operator is a symmetry operator that factorizes into mutually independent subsystem symmetry operators; its stratified anomaly is defined as the collection of anomalies associated with these subsystem operators. We develop a cellular chain complex formalism for stratified anomalies of internal symmetries and show that, in the presence of crystalline symmetries, they give rise to Lieb-Schultz-Mattis (LSM) constraints. This includes LSM anomalies and SPT-LSM theorems. We apply this framework to modulated $G$ symmetries, which are symmetries whose total symmetry group is ${G_\mathrm{tot} = G \rtimes G_\mathrm{s}}$, with $G_\mathrm{s}$ the crystalline symmetry group. Notably, a nonzero stratified anomaly within a fundamental domain of $G_\mathrm{s}$ (e.g., a unit cell) does not always imply an LSM anomaly for modulated symmetries. Instead, the existence of an LSM anomaly also depends on how $G_\mathrm{s}$ acts on $G$. When $G_\mathrm{s}$ is the lattice translation group, we find an explicit criterion for when a stratified anomaly causes an LSM anomaly, and classify LSM anomalies using homology groups of $G_\mathrm{s}$-invariant cellular chains. We illustrate this through examples of exponential and dipole symmetries with stratified anomalies, both in ${(1+1)}$D and ${(2+1)}$D, and construct a stabilizer code model of a modulated SPT subject to an SPT-LSM theorem.
Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions
讨论了Lieb-Schultz-Mattis定理在大于1的维数上的推广。本文解释了为什么前人所建立的变分波函数在一维以上的情况下对定理的证明没有帮助。短程R. V. B Sutherland，Rokhsar和Kivelson，Read和Chakraborty的图像给出了一个强有力的支持断言，该定理确实是有效的，在任何层面。一般观念的一些例证显示在精确的光谱上。
Non-invertible translation from Lieb-Schultz-Mattis anomaly
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN（理化学研究所跨学科理论与数学科学项目）
Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）；Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）
Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN（理化学研究所跨学科理论与数学科学项目）；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Center for Quantum Mathematics at IMADA, Southern Denmark University（南丹麦大学IMADA量子数学中心）；Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所尼尔斯·玻尔国际学院）
Lieb-Schultz-Mattis Constraints for Quantum Channels: A Spacetime-Duality View
Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Maryland, College Park（马里兰大学帕克分校）；University of Oklahoma（俄克拉荷马大学）；Technical University of Munich（慕尼黑工业大学）；Northeastern University（东北大学）
Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints
Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong（香港中文大学）；RIKEN（理化学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Stanford University（斯坦福大学）；University of Southern California（南加州大学）
Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
Dalhousie University（达尔豪斯大学）；Perimeter Institute for Theoretical Physics（珀蒂默理论物理研究所）；University of Waterloo（滑铁卢大学）；National University of Singapore（新加坡国立大学）
State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University（清华大学物理系低维量子物理国家重点实验室）；Qiuzhen College, Tsinghua University（清华大学邱成通数学科学中心）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Hefei National Laboratory（合肥国家实验室）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit
University of Granada（格拉纳达大学）；Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada（格拉纳达大学理论计算物理卡洛斯一世研究所）；University CEU Fernando III（塞乌费尔南多三世大学）；Higher Polytechnic School, University CEU Fernando III（塞乌费尔南多三世大学高等理工学院）；University of Jaen（哈恩大学）；Department of Mathematics, University of Jaen（哈恩大学数学系）
