晶体相互作用拓扑相与等变上同调：假设还是不假设
Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
摘要：
对于同构晶体相互作用带隙系统，我们导出了绝热演化下的分类。这种分类是完整的非简并基态。对于退化情形，我们讨论了由等变特征类给出的一些不变量。我们并不假定一个突现的相对论性场论，也不假定相位形成一个拓扑谱。我们也不局限于系统的短程纠缠，稳定性与平凡的系统堆叠，也不假设存在准粒子的SPT和SET分类分别完成。使用稍微广义的Bloch分解和Grassmanians的基态空间，我们表明，$P$等变上同调的$d$维环面引起不同的相互作用相，其中$P$表示的点群的晶体结构。我们比较我们的结果玻色子同形晶体SPT阶段和非相互作用的费米晶体阶段在A类。最后，我们讨论了我们的假设的关系，为结晶SPT和SET阶段。
英文摘要：
For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional torus gives rise to different interacting phases, where $P$ denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.
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