Classifying quantum phases using Matrix Product States and PEPS
Norbert Schuch, David Perez-Garcia, Ignacio Cirac
备注：
v2: 21 pages, 6 figures. Significantly rewritten and extended. Now contains a classification of phases both without and with symmetries, for systems with both unique and degenerate ground states. v3: 24 pages. Improved according to referees' suggestions (most notably added examples and improved definition of phases under symmetries). Accepted at Phys. Rev. B
在矩阵乘积态（MPS）及其相关的父哈密顿量的框架下，对具有唯一和简并基态的一维系统，在对称性存在和不存在的情况下，给出了带隙量子相位的分类.我们发现，没有对称性，所有的系统是在同一个阶段，直到偶然的基态简并。如果强加对称性，则没有对称性破坏的相位（即，具有唯一基态）按对称群的上同调类，即其射影表示的等价类进行分类，这是[X. Chen，Z. - C. Gu和X。- G. Wen，Phys.Rev.B83，035107（2011）; arXiv：1008.3745]。对于具有对称性破坏的相位（即，简并基态），我们发现对称性由两部分组成，一部分作用于基态的置换作用，另一部分作用于单个基态，相位由前者的置换作用和后者的上同调类来标记.使用投影纠缠对状态（PEPS），我们随后扩展我们的框架，在一些重要的情况下，在特定的系统具有独特的基态，退化的基态与当地的序参量，和拓扑顺序的附近的二维相位的分类。我们还表明，在两个维度上，施加对称性并不约束的相图在一维的方式。作为一个中心工具，我们介绍了等距形式，规范形式的MPS和PEPS，这是一个重整化不动点。将一个状态变换为它的等距形式并不改变相位，因此，我们可以把重点放在等距形式的分类上。
英文摘要：
We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.
Supervised Learning with Projected Entangled Pair States
Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Songshan Lake Materials Laboratory（松山湖材料实验室）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
Bond dimension witnesses and the structure of homogeneous matrix product states
在过去的二十年里，矩阵乘积态（MPS）被广泛地应用于固体物理中，用来近似一维自旋链的基态。在本文中，我们研究齐次MPS（hMPS），或MPS通过站点无关张量和边界条件。利用矩阵代数理论，我们得到了所有hMPS共有的两个结构性质：a）存在使给定键维数的所有hMPS零化的局部算子;和b）存在局部算子，当应用于给定键维数的任何hMPS时，将颗粒从自旋链上分离（切割），同时将两个松散的末端再次连接（粘合）成hMPS。有了这些工具，我们将展示如何系统地推导出“键维的”，或2-本地运营商的期望值，使我们能够降低键维的基础hMPS。我们将其中的一些结果推广到了投影纠缠对态（PEPS）的分析。作为奖励，我们使用我们对hMPS结构的洞察力：a）推导出使用hMPS和hPEPS进行基态能量计算的一些理论限制; b）显示如何降低复杂性并提高[Phys. Rev. Lett. 115，020501（2015）]用于表征有限维量子相关性。
A Classifying Space for Phases of Matrix Product States
University of California, Davis（加州大学戴维斯分校）；Center for Quantum Mathematics and Physics, University of California, Davis（加州大学戴维斯分校量子数学与物理中心）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Center for Theory of Quantum Matter, University of Colorado Boulder（科罗拉多大学博尔德分校量子物质理论中心）；University of Chicago（芝加哥大学）；Kadanoff Center for Theoretical Physics, University of Chicago（芝加哥大学卡达诺夫理论物理中心）；California Institute of Technology（加州理工学院）；Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院物理系与量子信息与物质研究所）
The local characterization of global tensor network eigenstates
Instituto de Física Teórica, UAM-CSIC（理论物理研究所，马德里自治大学-西班牙国家研究委员会）；University of Vienna, Faculty of Mathematics（维也纳大学数学学院）；University of Vienna, Faculty of Physics（维也纳大学物理学院）；Ghent University（根特大学）；University of Cambridge（剑桥大学）
