在拓扑性近藤绝缘体如SmB6和非中心对称性重费米子超导体如CePt3Si中，自旋-轨道耦合可能导致拓扑型物相的出现，并可能产生特殊的相变过程。基于项目组在重费米子领域的前期工作基础，本项目拟采用平均场理论、变分蒙特卡罗方法、量子蒙特卡罗模拟等方法，利用Anderson晶格模型和近藤晶格模型，首先研究在自旋-轨道耦合的作用下，拓扑性近藤绝缘体中的近藤效应、物相及表面态的拓扑性质，并研究随体系参数变化下的相变过程，解释实验观测到的磁性相变和拓扑相变；其次研究在自旋-轨道耦合的影响下，非中心对称性重费米子超导体的微观超导配对机制及其配对对称性，并研究磁性和超导的相互作用过程，给出包含反铁磁相和超导相及其共存相的相图。本项目预期将提升对拓扑性近藤绝缘体和非中心对称性重费米子超导体中的物相、相变以及拓扑性质的理论理解，并对相关材料的实验研究做出一定的理论参考。

In topological Kondo insulators (such as SmB6) and non-centrosymmetric heavy-fermions superconductors (such as CePt3Si), the spin-orbit coupling may induce non-trivial topological phases and unique quantum phase transitions. Based on previous works on heavy-fermion systems, we first adapt the mean-field theory, variational Monte Carlo method and quantum Monte Carlo simulation on the Anderson lattice model and Kondo lattice model, to study the Kondo effect, the surface states, the phase evolutions and phase transitions arising from the spin-orbit coupling in topological Kondo insulators, focusing on the experimental observed magnetic phase transition and topological phase transition. We also study the superconducting paring mechanism and paring symmetry of the non-centrosymmetric heavy-fermion superconductors in the context of spin-orbit coupling, then investigate the interplay between superconductivity and magnetism, to draw the phase diagram among the antiferromagnetic phase, superconducting phase and possible their coexisting phase. This project may improve the theoretical understanding of the phases, phase transitions and topological character induced by the spin-orbit coupling in topological Kondo insulator and non-centrosymmetric heavy-fermion superconductors, and may provide deeper supports to future experimental exploration in related materials.