电子间相互作用是凝聚态物理领域中许多新奇现象的根源。在电子关联体系中，自旋、电荷、轨道和晶格自由度之间相互影响，多个自由度的强烈耦合造成了复杂多变的物理性质。作为最简单的相互作用模型之一，Hubbard模型能够描述诸多关联电子体系的物理现象，包括Mott相变、磁序转变以及非常规超导。该模型承载了关联电子体系的本质，包含载流子的在位排斥作用和格点间跃迁，同时能够探究其他条件如掺杂、无序、外场等的改变对物理性质的影响。在理论和实验研究中，晶格系统内不同的载流子浓度与无序受到了广泛关注，因此本文基于Hubbard模型研究了二维关联系统中掺杂和无序对输运性质和磁性的影响。
  论文的第一章从关联电子体系的定义出发，讨论了该领域相关的物理现象。随后我们简要介绍了文章主要研究的二维狄拉克费米子体系，并在第二章中介绍了采用的模型和模拟方法，即Hubbard模型、行列式量子蒙特卡罗方法(DQMC)和约束路径量子蒙特卡罗方法(CPMC)。两种方法主要用于研究相互作用费米子体系，其中后者属于在可控近似下求解相互作用体系。正方晶格是最简单的晶格结构之一，能够模拟铜酸盐中d波超导电性和反铁磁涨落，一直以来都是引人关注的晶格体系。
  在第三章中，基于排斥Hubbard模型，我们利用DQMC方法研究了无序和掺杂对正方晶格中金属-绝缘体转变的影响，其中掺杂和键无序分别通过调节化学势和跃迁项引入。当系统偏离半满时，粒子-空穴对称性被破坏，DQMC方法的负符号问题会增大数值模拟的难度。因此，我们展示了负符号问题随各参数的变化情况，并选择合适的电子浓度、温度、键无序、在位排斥作用和晶格尺寸以保证模拟结果的可靠性。研究发现，向掺杂体系中引入键无序将诱发金属-绝缘体转变，并且临界无序强度会随电子浓度改变。我们通过二维相图反映了不同相互作用下二者对输运性质的影响，阐述了关联电子系统中复杂丰富的性质。
  Lieb晶格的原子排布与正方晶格类似，不同之处在于具有费米能级处完全平坦的能带结构。因此，Lieb晶格具有诸多不同于正方晶格的物理性质。在第四章中，我们利用DQMC方法研究了掺杂对无序Hubbard模型下Lieb晶格的影响，并通过选择合适的电子浓度减弱了符号问题的影响，使结果更加可靠。研究发现，相较于半满下Lieb晶格始终表现为绝缘性，掺杂系统中相互作用与无序彼此影响诱发了金属-绝缘体转变。其中，无序局域化电子、相互作用离域电子，二者呈现竞争关系。此外，无论电子是否填充平带，相同条件下Lieb晶格电导率始终小于正方晶格，代表平带的存在可能抑制金属性。
  稀释(dilution)是一种用非磁性离子代替磁性离子的无序。应用DQMC方法，在第五章工作中，通过将独立于格点的在位排斥作用随机设置为0，我们向基于Hubbard模型的六角晶格系统引入稀释。根据电导率、态密度和反铁磁结构因子的结果，我们揭示了一种由稀释和排斥竞争引起的新奇中间绝缘相。随着非磁性离子掺杂程度的增加，这种非磁性中间相出现在分离金属和Mott绝缘相的零温量子临界点，并且具有鲁棒性。基于强关联材料，我们认为通过掺杂非磁性离子，可以有效地将六角晶格体系中的Mott绝缘相转化为顺磁金属相。这一结果不仅与实验研究中稀释影响磁序的结论一致，而且为相关材料中金属-绝缘体转变的研究提供可能的方向。
  最后我们讨论了与六角晶格具有相似能带结构的交错通量正方晶格。利用DQMC方法，我们分别计算了干净体系中的电导率和反铁磁自旋结构因子，发现随着相互作用强度增加，金属-绝缘体转变和反磁转变同时发生，而施加无序在加速金属-绝缘体转变的同时会抑制反铁磁转变，因此系统出现中间相。大量研究证实了载流子掺杂有利于超导配对的产生，因此我们运用CPMC方法研究了掺杂交错通量正方晶格中各类对称性的关联函数，发现dxy波占据主导地位，并且非局域相互作用不会改变系统的主导对称性。我们的研究可能为狄拉克费米子系统的超导电性提供新视野。

  Electron-electron interaction is considered as origin of many novel phenomena in condensedmatter physics. In the electron correlated system, the degrees of freedom of spin, charge, orbital and lattice interact with each other, and the strong coupling of multiple degrees of freedom results in complex and diverse physical properties. As one of the simplest models to describe correlated fermions, the Hubbard model is able to probe many interesting problems, including Mott transition, magnetic order transition, and unconventional superconductivity. This model carries the essence of the correlated electron system, including the on-site repulsion of carriers and inter-site transition. It allows researchers to explore the influence of other conditions on physical properties, such as doping, disorder and external field. In theoretical and experimental studies, different carrier concentrations and disorder in lattice system have received extensive attention. Therefore, based on the Hubbard model, in this thesis we study the effect of doping and disorder on transport properties and magnetism in two-dimensional correlated systems.
  In the first chapter of this thesis, we start from the definition of correlated electron system and discuss the novel phenomena in related fields. Then we briefly introduce the characteristics of two-dimensional Dirac Fermion systems. While for the Chapter II, the Hubbard model and two simulation methods are discussed in this part, including the Determinant Quantum Monte Carlo method (DQMC) and the Constrained Path quantum Monte Carlo method (CPMC). Both methods are applied to study the interacting fermion systems, and the latter one solves systems under the
controllable approximation.
  As a simple model of simulating d-wave pairing superconductivity and antiferromagnetic fluctuations of cuprate, the square lattice has attracted much interest. In chapter III, we use DQMC method to examine the effect of disorder and doping on the metal-insulator transition in a repulsive Hubbard model on a square lattice. Doping and bond disorder are introduced respectively by adjusting the chemical potential and hopping term. When the system deviates from half-filling, the particle-hole symmetry is destroyed and DQMC method suffers from sign problem. To make our results reliable, we compute the sign problem of system under various parameters, and select appropriate parameters such as electron density, temperature, bond disorder, on-site interaction, and lattice size. We show that randomness in the hopping elements induces the metal-insulator transition and the critical disorder strength  differs at different fillings. We reflect their effects on the transport properties under different interactions by plotting two-dimensional phase diagram, and show the complex and rich properties of correlated electron system.
  Although the Lieb lattice has an atomic arrangement similar to that of the square lattice, it hosts many different novel properties due to its flat band centered at the Fermi level. In Chapter IV, we use DQMC method to investigate the influence of doping on the Lieb lattice under the disordered Hubbard model. By choosing suitable electron densities, we ensure the weak sign problem to improve the reliability of our results. It is found that disorder and on-site repulsive interaction induce the metal-insulator transition which is impossible in the half-filled case. Among them, disorder localizes electrons while interaction plays the opposite role, showing a competitive relationship. Furthermore, the conductivity of Lieb lattice is always smaller than that of square lattice under the same conditions, suggesting that the flat band suppresses the metallicity whether it is filled with electron or not.
  Dilution is a kind of disorder in which magnetic ions are replaced by nonmagnetic ions. In Chapter V, using DQMC calculation, we investigated an interacting Dirac fermion model with the on-site repulsion being randomly zero on a fraction x of sites. Based on the conductivity, density of states at the Fermi energy and antiferromagnetic structure factor, our results reveal a novel intermediate insulating phase induced by the competition between dilution and repulsion. With increasing doping level of nonmagnetic ions, this nonmagnetic intermediate phase is found to emerge from the zero temperature quantum critical point separating a metallic and a Mott insulating phase, whose robustness is proven over a wide range of interactions. Under the premise of strongly correlated materials, we suggest that doping nonmagnetic ions can effectively convert the system back to the paramagnetic metallic phase. This result not only agrees with experiments on the effect of dilution on magnetic order but also provides a possible direction for studies focusing on the metal-insulator transition in honeycomb lattice-like materials.
  Finally, we discuss the Π−flux Hubbard model on a square lattice whose energy band structure is similar to the one of honeycomb lattice. Using DQMC method, we calculated the conductivity and antiferromagnetic structure factor in the clean system. It is found that the metal-insulator transition and antiferromagnetic phase transition occur simultaneously with increasing interaction strength. The addition of disorder accelerates the metal-insulator transition while suppresses the antiferromagnetic phase transition, thus the introduction of disorder causes an intermediate phase in the system. With many literatures confirming that carrier doping is beneficial to the generation of superconducting pairings, we apply CPMC method to compare pairing correlation function of different pairing symmetry. The dxy wave is found to dominate in doping system, and its dominance cannot be changed by the non-local interaction. Our numerical results may provide some useful insight on the understanding of interaction driven superconductivity in Dirac Fermion systems.