平方费米子：三重方法和零模的命运
Squaring the fermion: The threefold way and the fate of zero modes
摘要：
研究了稳定玻色子系统的平均场论的拓扑性质和分类。在三种标准的对称性分类中，只有时间反演代表了多玻色子系统的真实对称性，而另外两种，粒子-空穴和手征，只是表现为有效单粒子问题对称性的约束。对于任意空间维度的带隙系统，我们建立了三个基本的不去定理，证明了不存在：奇偶开关，保序拓扑量子相位，和开边界条件下的局部玻色子零模。然后，我们引入了一个平方，内核保持地图连接非相互作用的厄米理论的费米子和稳定的玻色子系统，它作为一个游乐场，以揭示拓扑结构的作用，在玻色子阶段和本地化的midgap边界模式。最后，我们确定继承的费米十倍的方式分类的对称类，揭示了一个优雅的三倍的方式拓扑分类的非相互作用玻色子。我们说明了我们的主要研究结果在一维和二维玻色子晶格和场论模型。
英文摘要：
We investigate topological properties and classification of mean-field theories of stable bosonic systems. Of the three standard classifying symmetries, only time-reversal represents a real symmetry of the many-boson system, while the other two, particle-hole and chiral, are simply constraints that manifest as symmetries of the effective single-particle problem. For gapped systems in arbitrary space dimension we establish three fundamental no-go theorems that prove the absence of: parity switches, symmetry-protected-topological quantum phases, and localized bosonic zero modes under open boundary conditions. We then introduce a squaring, kernel-preserving map connecting non-interacting Hermitian theories of fermions and stable boson systems, which serves as a playground to reveal the role of topology in bosonic phases and their localized midgap boundary modes. Finally, we determine the symmetry classes inherited from the fermionic tenfold-way classification, unveiling an elegant threefold-way topological classification of non-interacting bosons. We illustrate our main findings in one- and two-dimensional bosonic lattice and field-theory models.
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