[论文解读] Fractional Chern Insulator

Nicolas Regnault, B. Andrei Bernevig|arXiv (Cornell University)|May 24, 2011

Topological Materials and Phenomena参考文献 21被引用 182

一句话总结

该论文通过在平坦能带极限下对Hubbard相互作用进行对角化，为在无外加磁场的陈绝缘体中于1/3填充时出现分数量子霍尔效应（FQHE）提供了确凿的数值证据。该系统表现出三重简并的基态，具有有限能隙、不可压缩性、动量空间中1/3的恒定密度，以及与Laughlin准-hole态计数相符的纠缠谱——强烈表明存在分数量子陈绝缘体相。

ABSTRACT

Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at 1/3 filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground state density is constant, and equal to 1/3 in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator - both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.

研究动机与目标

• 在无朗道能级的拓扑绝缘体中确立分数量子霍尔态的存在。
• 将分数量子陈绝缘体与竞争性的电荷密度波态区分开来。
• 通过不可压缩性、简并性及分数统计，证明基态表现出拓扑序。
• 确认Laughlin型准-hole激发态的出现以及鲁棒的纠缠能隙。

提出的方法

• 在棋盘格陈绝缘体模型的平坦能带极限下，对Hubbard相互作用进行精确对角化。
• 采用广义泡利不相容原理，禁止在三个连续轨道上存在超过一个粒子。
• 计算动量空间密度以排除电荷密度波序。
• 通过磁通插入下的能谱流分析，确认拓扑简并性。
• 计算纠缠谱并识别出与Laughlin准-hole态计数匹配的有限纠缠能隙。
• 研究系统在单粒子哈密顿量相变下的响应，对比拓扑与平凡绝缘体相区。

实验结果

研究问题

• RQ1 在无外加磁场的陈绝缘体中，于1/3填充时能否出现分数量子霍尔态？
• RQ2 如何将分数量子陈绝缘体与竞争性的电荷密度波态区分开来？
• RQ3 多体基态是否表现出拓扑序，包括分数统计与不可压缩性？
• RQ4 基态的纠缠谱是否与Laughlin态一致？
• RQ5 当系统经历拓扑相变时，纠缠能隙在热力学极限下是否仍保持有限？

主要发现

• 在1/3填充时，系统表现出三重简并的基态，其有限能隙在热力学极限下依然存在。
• 动量空间密度均匀且等于1/3，排除了电荷密度波序的可能性。
• 磁通插入下的能谱流证实了拓扑简并性，且无能级排斥现象。
• 纠缠谱显示出清晰的能隙，能隙以下的态计数与Laughlin准-hole态一致。
• 纠缠能隙在拓扑相变过程中保持有限，并与多体能隙同步演化。
• 激发态的计数与Laughlin准-hole态一致，证实了分数统计与拓扑序。