晶格中的半周期相位：边缘态、纠缠谱和平带
Haldane phase in the sawtooth lattice: Edge states, entanglement spectrum and the flat band
摘要：
利用密度矩阵重整化群数值计算方法，研究了在动能项下具有强阻挫的双晶格中半满玻色-哈伯德体系的相图.我们特别关注的跳跃项产生一个平坦的频带的值，并表明，在接触和近邻排斥的存在下，存在三个阶段：莫特绝缘体（MI），电荷密度波（CDW），和拓扑Halperson绝缘（HI）相，显示系统的两端之间的边缘状态和粒子的不平衡。我们发现，即使纠缠谱在Halfton阶段是不是双重退化，它是在很好的协议与阿弗莱克-肯尼迪-Lieb-Tasaki（AKLT）的纠缠谱建立在Wannier基与平带。这强调了纠缠谱中没有简并不一定是非拓扑相位的特征，而是（隐藏的）保护对称性涉及非局域态。最后，我们还表明，HI阶段是稳定的，对小偏离平坦的频带，但被破坏为较大的。
英文摘要：
Using density matrix renormalization group numerical calculations, we study the phase diagram of the half filled Bose-Hubbard system in the sawtooth lattice with strong frustration in the kinetic energy term. We focus in particular on values of the hopping terms which produce a flat band and show that, in the presence of contact and near neighbor repulsion, three phases exist: Mott insulator (MI), charge density wave (CDW), and the topological Haldane insulating (HI) phase which displays edge states and particle imbalance between the two ends of the system. We find that, even though the entanglement spectrum in the Haldane phase is not doubly degenerate, it is in excellent agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state built in the Wannier basis associated with the flat band. This emphasizes that the absence of degeneracy in the entanglement spectrum is not necessarily a signature of a non-topological phase, but rather that the (hidden) protecting symmetry involves non-local states. Finally, we also show that the HI phase is stable against small departure from flatness of the band but is destroyed for larger ones.
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