电子间相互作用一直是凝聚态物理领域新颖物理现象的研究重点，它被广泛认为是揭示磁性和非常规超导电性内在机理的关键因素。在强关联电子体系中，电荷、自旋等多个自由度之间相互作用，导致体系的物理性质具有多样性和复杂性。Hubbard 模型是一种描述晶格系统中关联效应的基本模型，为强关联体系的研究提供了理论支撑。该模型包含了库仑相互作用和格点间跃迁作用，用来探究关联电子系统所关心的诸多物理性质，包括磁关联、非常规超导、以及超导转变温度等。棋盘晶格作为阻挫结构的同时具备平带特征，实验和理论上都被视 作有趣的晶格系统，其包含的各种物理性质受到了广泛的关注。因此本文的研究基于棋盘晶格 Hubbard 模型，通过引入次近邻跃迁相互作用调控阻挫，利用数值方法分析了阻挫强度和掺杂对磁性和超导配对的影响。

本论文主要工作有：利用行列式量子蒙特卡罗方法 (DQMC)，在电子掺杂棋盘晶格的 Hubbard 模型参数空间中，分析了在位相互作用、温度、电子浓度以及阻挫强度对自旋磁化率的影响。研究表明，对于不同的阻挫强度，在电子浓度 ⟨n⟩ ≳ 1.2 的电子掺杂区域，体系均表现出稳定的强铁磁涨落特性，并且随着相互作用增强和温度降低，自旋磁化率显著增强。同时，我们还评估了负符号问题， 以明确哪个参数区域是可以有效且可靠地进行模拟计算。我们的结果不仅对调控棋盘格晶格中的磁性具有重要意义，还为探索平带系统中丰富的关联行为提供了研究思路。

我们利用 DQMC 方法研究了空穴掺杂棋盘晶格中的超导配对，确定了棋盘晶格 Hubbard 模型的主要配对对称性。研究结果表明，在阻挫强度较大的区域，向空穴掺杂棋盘晶格中引入次近邻跃迁相互作用诱发了配对对称性改变，且不改变反铁磁涨落，同时配对对称性会随电子浓度改变。我们通过二维相图描述了d波和 d+ is波在电子浓度和阻挫强度参数空间中的范围，阐述了关联电子系统中阻挫和超导的关系。

我们利用约束路径量子蒙特卡罗方法 (CPMC)，有效地避开负符号问题，研 究了定义在掺杂棋盘晶格 Hubbard 模型中的超导配对关联，其中掺杂和阻挫强度分别由电子填充和次近邻跃迁项调控。我们发现：在阻挫强度较小的空穴掺杂棋盘晶格中，d波对称性的超导配对关联函数优于其他对称性；d波配对关联函数随着次近邻跃迁强度的减小而增大; Vertex 函数反映的是相互作用产生的效果，随着库仑相互作用的增加而增大。研究结果为进一步理解超导体中的关联效应及其超导配对机制提供了有价值的参考。

基于吸引相互作用 Hubbard 模型，我们利用 DQMC 方法数值模拟晶格尺寸 最大至L= 14，主要计算了棋盘晶格的超导配对关联。研究发现，s波超导配对关联随温度和尺寸的变化显示出超导相变的特征，利用有限尺寸标度理论的分析，我们可以从中提取超导转变温度。此外，本工作还利用超流密度在相变温度 的普适跃迁关系确定了转变温度。结合两种数据分析方法说明了结果的可靠性。 我们注意到，棋盘晶格中超导的变化趋势与在部分三角晶格化合物以及 Kagome 化合物中所观察到的超导现象类似，具有进一步研究的价值，为展现丰富的强关联现象提供了平台。

Electron-electron interactions have long been a central topic in the study of novel physical phenomena within the realm of condensed matter physics, widely recognized as a key factor in unraveling the underlying mechanisms of magnetism and unconventional superconductivity. In strongly correlated electronic systems, the interplay among multiple degrees of freedom such as charge and spin leads to a diverse and complex array of material properties. Hubbard model is a fundamental model for describing the correlation effects in lattice systems, providing theoretical support for the study of strongly correlated systems. This model encompasses both Coulomb interactions between electrons and hopping processes between lattice sites, enabling the investigation of a multitude of relevant physical properties in correlated electronic systems, including magnetic correlations, unconventional superconductivity, and superconducting transition temperatures. The checkerboard lattice is an intriguing lattice system, characterized by its simultaneous manifestation of geometric frustration and flat-band features. Both experimentally and theoretically, it attracts widespread interest due to the diverse range of physical properties it possesses. Consequently, our present research is grounded in the checkerboard lattice Hubbard model, incorporating next-nearest neighbor hopping interactions as a means to tune frustration strength, and employs numerical techniques to examine the effects of frustration strength and doping on magnetic behavior and superconducting pairing.

The main work introduced in this thesis includes: Using the Determinant Quantum Monte Carlo(DQMC) method to analyze the influence of onsite interactions, temperature, electron density, and frustration strength on the spin magnetic susceptibility within the parameter space of the Hubbard model for an electron-doped checkerboard lattice. Our findings demonstrate that, for varying degrees of frustration, in the region of electron doping where the average electron concentration ⟨n⟩ ≳ 1.2, the system consistently displays robust ferromagnetic fluctuations. Moreover, the spin magnetic susceptibility is found to increase significantly as both the interactions intensify and the temperature decreases. Concurrently, we also address the issue of negative signs, identifying the parameter regime where effective and reliable simulations can be conducted. These results not only hold considerable importance for manipulating magnetism within checkerboard lattices but also provide a conceptual framework for investigating the rich correlated behaviors in flat-band systems.

We investigated superconducting pairing in hole-doped checkerboard lattices by using DQMC method, thereby identifying the primary pairing symmetries within the Hubbard model for checkerboard lattices. Our findings indicate that, in regions of large frustration, the introduction of next-nearest neighbor hopping interactions into hole doped checkerboard lattices induces a change in the pairing symmetry without altering the antiferromagnetic fluctuations, and the pairing symmetry also changes with the electron filling. We demonstrate the relationship between frustration and superconductivity in correlated electronic systems through a two-dimensional phase diagram, point out the regimes of d-wave and d + is wave pairing symmetries in the parameter space of electron filling and frustration strength.

We employed the Constrained Path Quantum Monte Carlo(CPMC) method to avoid the sign problem and investigate the superconducting pairing correlations defined within the doped checkerboard lattice Hubbard model, where doping and frustration strength are controlled by adjusting electron filling and modulating next-nearest neighbor hopping terms. Our findings reveal that, in lightly frustrated hole-doped checkerboard lattices, the superconducting pairing correlation functions with d-wave symmetry is dominant pairing symmetry over other pairing symmetries. The d-wave pairing correlation function is found to increase with decreasing next-nearest neighbor hopping strength. The vertex function reflects the effects generated by interactions, which increase with the enhancement of Coulomb interaction strength. These research provide valuable insights into the role of correlations in superconductors and their pairing mechanisms.

Based on the attractive interaction Hubbard model, we employ the DQMC method to perform numerical simulations up to a maximum lattice size of L = 14. We mainly focus on calculating the superconducting pairing correlations in the checkerboard lattice. Our study reveals that the ??-wave superconducting pairing correlations exhibit characteristic signatures of a superconducting phase transition as temperature and system size are varied. Through the utilize of finite-size scaling theory analysis, we are able to extract the superconducting transition temperature from these data. Additionally, this work utilizes the universal jump in superfluid density at the phase transition temperature to determine the transition temperature. The combination of two data analysis methods illustrates the reliability of the results. We note that the trend of superconductivity in the checkerboard lattice bears resemblance to the superconducting phenomena observed in certain triangular lattice compounds and Kagome compounds, suggesting it is worth further research, as it provides a platform for unveiling a rich array of strongly correlated phenomena.