[论文解读] Towards a mathematical formalism for classifying phases of matter

Andreas Bauer, Jens Eisert+1 more|arXiv (Cornell University)|Mar 13, 2019

Quantum many-body systems参考文献 81被引用 49

一句话总结

论文提出一种基于张量格和局部移动的统一数学框架，用于对物质相进行分类，涵盖对称性破缺、拓扑相及相关结构，并扩展到边界、缺陷，以及在2+1D中的anyons。

ABSTRACT

We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor networks to those lattices. Different lattices are related by local operations called moves. Those local operations define consistency conditions for the tensors of the tensor network, the solutions to which yield exactly solvable models for all kinds of phases. We implement the framework to obtain models for symmetry-breaking and topological phases in up to three space-time dimensions, their boundaries, defects, domain walls and symmetries, as well as their anyons for 2+1-dimensional systems. We also deliver ideas of how other kinds of phases, like SPT/SET, fermionic, free-fermionic, chiral, and critical phases, can be described within our framework. We also define another structure called contracted tensor lattices which generalize tensor lattices: The former associate tensors instead of tensor networks to lattices, and the consistency conditions for those tensors are defined by another kind of local operation called gluings. Using this generalization, our framework also covers mathematical structures like axiomatic (non-fully extended or defective) TQFTs, that do not directly describe phases on a microscopic physical level, but formalize certain aspects of potential phases, like the anyon statistics of 2+1-dimensional phases. We also introduce the very powerful concept of (contracted) tensor lattice mapping, unifying a lots of different operations, such as stacking, anyon fusion, anyon condensation, equivalence of different fixed point models, taking the Drinfel'd centre, trivial defects or interpreting a bosonic model as a fermionic model.

研究动机与目标

• 发展用于分类物质相的统一数学框架。
• 引入张量格作为格点上张量网络的局部处方。
• 定义将不同晶格相关联并对张量网络施加一致性约束的局部移动。
• 在最多3个时空维度的对称性破缺和拓扑相上演示该框架。
• 探索向边界、缺陷、域墙、对称性和任何子的扩展；概述SPT/SET、费米、手性和临界相的描述。

提出的方法

• 将张量格定义为将张量网络与晶格关联的局部处方。
• 通过将晶格相关联的局部移动来强制一致性并约束张量网络。
• 引入带有拼接的收缩张量格及相应的一致性概念。
• 展示该框架如何在2+1和3+1维中给出各种相的严格可解模型。
• 发展“(收缩的)张量格映射”以统一诸如堆叠、anyon融合/凝聚以及Drinfeld中心解释等操作。
• 与公理化TQFT相关联以形式化超越直接显微模型的某些相的要点。

实验结果

研究问题

• RQ1 单一的数学形式是否可以在不同维度上同时对称性破缺和拓扑相进行分类？
• RQ2 局部移动与一致性条件如何产生不同相的严格可解模型？
• RQ3 收缩张量格及拼接在包含更广泛的相结构（如TQFT、anyon统计）中的作用是什么？

主要发现

• 统一的张量-格框架可以描述到3个时空维度的对称性破缺和拓扑相。
• 局部移动定义晶格表征之间的等价关系并强制张量网络的一致性。
• 收缩张量格推广了张量格并与拼接相连接，从而实现对边界、缺陷和anyon的描述。
• 该方法为在同一形式化框架内描述SPT/SET、费米、自由费米、手性和临界相提供途径。
• 与公理化TQFT的连接被确立，形式化了如2+1D中的anyon统计等方面。
• 张量-格映射统一了堆叠、anyon融合、凝聚和不动点模型等操作。