Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
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DOI：
10.1103/physrevresearch.2.043033
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发表时间：
2020-10-06
影响因子：
4.2
通讯作者：
Barkeshli, Maissam
中科院分区：
文献类型：
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作者：
Bulmash, Daniel;Barkeshli, Maissam
在（2+1）维[（2+1）D]拓扑有序的物质相中，某些对称性分形模式可能是反常的，这意味着它们对在纯（2+1）D中实现具有障碍。在本文中，我们演示了如何计算玻色子的富熵拓扑态的异常在完全的一般性。我们演示了如何给定全局对称群G的任何酉模张量范畴（UMTC）和对称分解类，定义一个（3+1）维[（3+1）D]拓扑不变的路径积分.我们提出了一个精确可解的哈密顿系统，并明确证明了（2+1）D G-对称表面终止主机deconfined任意子激发所描述的给定的UMTC和对称分形类。我们提出了具体的算法，可用于计算异常指标一般。我们的方法适用于一般的对称群，包括任意子置换和反酉对称。除了提供一种计算反常的一般方法外，我们的结果还表明，通过显式构造，任何UMTC的每一个对称性分数化类都可以在（3+1）维SPT态的表面实现.作为一个副产品，这种构造还提供了一种方法，可以明确地看到定义对称性分解的代数数据通常是如何在精确可解模型的上下文中出现的。在酉方向保持对称性的情况下，我们的结果也可以被看作是提供了一种方法来计算H-4（G，U（1））的障碍，出现在理论的G-交叉编织张量范畴，没有一般的方法已经提出了日期。
Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a G-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D G-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H-4(G, U(1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date.