低维材料具有丰富的物理性质，例如量子限制效应、态密度的变化、相位相干、弹道输运、弱局域化等等。在本文中我们主要研究低维材料的拓扑性质和光学性质。低维材料有限的尺寸可以对拓扑性质进行调控。我们关心的材料是最近发现的第一个由非点式对称性保护的拓扑材料KHgSb。它在(010)表面具有漏斗型费米子，且具有spectral flow的行为。我们通过第一性原理计算发现，KHgSb (010)薄膜具有和其(010)表面不同的能带连接方式。我们对KHgSb (010)方向的薄膜进行了细致的群表示论和相容关系的分析，发现有限尺寸效应确实可以破坏其漏斗型的能带连接方式。同时，我们也发现在某些情况下非点式对称性可以保护漏斗型费米子使其不受有限尺寸效应的影响。此外，第一性原理计算的结果还显示了漏斗型费米子具有特殊的自旋结构。用光场调控材料的拓扑性质的手段通常称为Floquet工程。在光场下，系统处于非平衡态，可能实现平衡态所不具有的特殊的性质。我们研究了磁掺杂拓扑绝缘体薄膜在光场下拓扑性质和磁性质的变化。我们发现随着光场的增强，拓扑绝缘体薄膜可以经历若干个拓扑相变过程，相变过程的数目取决于薄膜的厚度。更重要的是，我们发现光场还可以使电子磁化率下降，从而导致磁性薄膜出现铁磁到顺磁的磁相变。最后，我们考虑了Floquet体系在费米热库存在时的响应。为了描述具有热库的非平衡体系，我们首先引入了Floquet格林函数，使其可以通过Keldysh围道方法求解。我们用Floquet格林函数给出了Floquet体系的一阶响应的公式，并验证了它与平衡态响应函数的一致性。最后我们将Floquet响应理论应用在单层WS$_2$的磁化率计算中，发现在光场下WS$_2$的磁化率增强。其机制还需进一步地探讨。通过上述探索，我们希望我们的工作可以有助于人们理解光与物质的相互作用。
Low dimensional materials tend to have rich physical properties, such as, quantum confinement effects, the change of density of states, phase coherence, ballistic transpotation and weak localization etc. In this work, we mainly focus on the topological and optical properties of low dimensional materials. As is well know, quantum confinement effects could tune the topological properties of low dimensional materials. The particular material we care about here is KHgSb. It is the first topological material found that is protected by a nonsymmorphic symmetry. It has hourglass fermions on its (010) surface, manifesting a spectral flow behaviour. By first principles calculations, we found very different behaviours for (010) thin films compared with (010) surface. We did a thorough analysis by group representation theory and compatibility relations and proved that the finite size effects can break the hourglass fermions in most cases. There are exceptions when an additional nonsymmorphic symmetry is present. We also found that the hourglass fermions can have exotic spin texture which is never reported before. Tuning topological properties by light is usually treated as an expample of Floquet engineering. Especially, light can induce nonequilibrium state and the properties of the system may be altered compared to its equilibrium configuration. We studied the topological and magnetic properties of magnetic doped topological insulators (MDTIs) under illumination. We found that as the light intensity increase, the MDTI thin films can go through several topological phase transitions (TPTs). The number of TPTs is determined by the thickness of the thin films. More importantly, we found that the illummination of a high frequency light can induce a decrease in electron magnetic susceptiblity, leading to a magnetic phase transition from ferromagnetic to paramagnetic. At last, we considered the Floquet system with fermionic resevoirs. To describe a nonequlibrium system like this, we first introduced the Floquet Green's function, which is solvable by Keldysh contour technique. We then developed the first order response theory based on the Floquet Green's function and testified its validity by checking the consistency with the equlibrium Kubo formula. Finally, we applied our Floquet response theory in the calculation of WS$_2$'s magnetic susceptibility. We found the susceptibility grows with the increase of the light intensity. The mechanism behind needs further investigation. Through the above examples, we hope our work could shed some light on the understanding of light-matter interaction.
