We consider a tight-binding model with the nearest neighbour hopping integrals on the honeycomb lattice in a magnetic field. Assuming one of the three hopping integrals, which we denote t_a, can take a different value from the two others, we study quantum phase structures controlled by the anisotropy of the honeycomb lattice.For weak and strong t_a regions, respectively, the Hall conductances are calculated algebraically by using the Diophantine equation. Except for a few specific gaps, we completely determine the Hall conductances in these two regions including those for subband gaps. In a weak magnetic field, it is found that the weak t_a region shows the unconventional quantization of the Hall conductance, σ_{xy}=-(e^2/h)(2n+1), (n=0,\pm 1,\pm 2,...), near the half-filling, while the strong t_a region shows only the conventional one, σ_{xy}=-(e^2/h)n,(n=0,1,2,...). From topological nature of the Hall conductance, the existence of gap closing points and quantum phase transitions in the intermediate t_a region are concluded. We also study numerically the quantum phase structure in detail, and find that even when t_a=1, namely in graphene case, the system is in the weak t_a phase except when the Fermi energy is located near the van Hove singularity or the lower and upper edges of the spectrum.
Quantum Hall conductance and de Haas van Alphen oscillation in a tight-binding model with electron and hole pockets for (TMTSF)$_2$NO$_3$
在（TMTSF）$_2$ NO $_3$的紧束缚模型中，由于负离子在$a$ -方向上的周期性有序势，导致了电子和空穴的小口袋，从理论上研究了量子化霍尔电导和de Haas van Alphen（dHvA）振荡.磁场被跳跃处理为复数，这是由于矢量位引起的相位，即Peierls替换。在参数和磁场的实际值中，获得作为磁场的函数的能量（Hofstadter蝶形图）。结果表明，作为反磁场的函数，能级加宽，带隙周期性地闭合或几乎闭合，这在磁击穿的半经典理论中是看不到的。当化学势处于能隙内时，霍尔电导被量子化为一个由丢番图方程得到的整数。当在该系统中掺杂电子或空穴时，霍尔电导在磁场的某些区域中被量子化，但由于朗道能级的加宽，霍尔电导在磁场的其它区域中不被量子化。零温下dHvA振荡的振幅随磁场的增大而减小，而在半经典的Lifshitz Kosevich公式中dHvA振荡的振幅是常数.
Topological quantum phase transition in Kane-Mele-Kondo lattice model
我们系统地探讨了蜂窝晶格上的Kane-Mele-Kondo晶格模型的基态相图，特别是，我们专注于它的磁性，这在以前的出版物[Feng，Dai，Chung，and Si，Phys. Rev. Lett. \textbf{111}，016402（2013）]。除了在那篇论文中发现的近藤绝缘体之外，还发现了两种反铁磁自旋密度波相。一个是正常的反铁磁自旋密度波态，另一个是具有量子化自旋霍尔电导和螺旋边缘态的非平凡拓扑反铁磁自旋密度波态。量子自旋霍尔绝缘体被发现是不存在的，因为它总是不稳定的反铁磁自旋密度波状态，至少在我们的模型中的平均场水平。此外，两个自旋密度波相之间的跃迁是由三维量子电动力学描述的拓扑量子相变，其中传导电子对低能Dirac费米子有贡献，而局域自旋的自旋波涨落产生有效的动态U（1）规范场.这种非平凡的转变表现出自由基临界热力学、输运和单粒子行为，为这种转变提供了一个指纹。此外，反铁磁自旋密度波态到近藤绝缘体的跃迁是一阶的。两个新的磁相的引入及其拓扑量子相变显示了Kane-Mele-Kondo晶格模型中包含的丰富的内在物理。
