铜氧化物高温超导材料的发现为凝聚态理论研究带来了巨大的挑战，其母体化合物为反铁磁莫特绝缘体，因此体系中强关联效应受到了凝聚态物理领域学者们的广泛关注。过渡金属氧化物中d轨道的高简并度使得多轨道效应成为强关联电子体系中一个重要的问题，2019 年发现的新型铜氧化物超导材料Ba₂CuO_4−δ 具有显著的双轨道特性，这与用Zhang-Rice单轨道模型所描述的传统铜氧化物高温超导材料存在差异，因此探究关联效应对于双轨道强关联体系中的量子相变以及Ba₂CuO_4−δ电子结构的影响具有重要意义。本论文基于以Lanczos方法作为杂质求解器的动力学平均场理论对双轨道强关联体系进行了研究，我们首先从双轨道Hubbard 模型出发，研究了电子关联效应以及带宽比对不同晶格体系中轨道选择莫特相变的影响；然后我们基于密度泛函理论所得的Ba₂CuO_4−δ能带结构，研究了不同电子关联效应下体系电子态的变化；最后我们通过计算双轨道Hubbard 模型中的超导序参量，研究了带宽存在差异时体系中超导态的轨道选择特性。
在对论文研究背景以及动力学平均场理论进行了系统介绍之后，我们首先研究了双轨道Hubbard模型中的轨道选择莫特相变。洪特耦合J通过抑制轨道间电荷的涨落来促进双轨道体系的轨道选择莫特相变，这种效应在J ∼ 0.25U时达到饱和；同时不同带宽的轨道上电荷受相互作用的影响也不一致，因此两轨道带宽差异越大体系越容易发生轨道选择莫特相变。我们在双轨道模型中引入轨道间跃迁，通过正则变换消除轨道间的杂化并得到退耦合的有效轨道。我们发现轨道间跃迁通过调控有效轨道的跃迁系数进而促进退耦和的有效双轨道模型中的轨道选择莫特相变。最后我们对比了正方晶格和Bethe 晶格中的轨道选择莫特相变，二者物理图像一致但相变点存在数值上差异，这主要是由于两种晶格能带宽度的差异，若统一它们的能带宽度，则相变点的差异将显著减小，因此Bethe 晶格被广泛应用于动力学平均场理论的研究之中。
我们将以Lanczos方法为杂质求解器的动力学平均场理论进行了拓展，使其能够处理含各向异性轨道间跃迁的能带结构，并根据密度泛函理论得到的Ba₂CuO_4−δ的能带结构研究了电子关联以及掺杂效应对Ba₂CuO_4−δ量子相变的影响。我们发现e_g轨道半填充的Ba₂CuO_3.5在特定的相互作用下会呈现轨道选择莫特相的特性，而Ba₂CuO_4−δ属于过渡金属氧化物材料，体系中具有一定强度的电子关联效应，因此处于轨道选择莫特相的Ba₂CuO_3.5有可能是Ba₂CuO_4−δ超导材料的母体化合物。在强洪特耦合作用下，随着e_g轨道电子填充数的增多，Ba₂CuO_4−δ体系先呈现出显著的轨道选择莫特相特征，而后转变为金属相，根据体系态密度我们发现此时体系应处于轨道选择莫特相的边缘态。在观测到超导效应的Ba₂CuO_3.2中，两e_g轨道的电子态在相互作用变化时呈现出不同特征，同时各轨道电子占据数的图像说明体系具有明显的双轨道特性。在体系处于轨道选择莫特相时，体系中具有极其稳定的轨道极化，我们据此提出了多轨道体系中可能存在的一种超导机制：一个或者多个轨道的电子为超导提供自旋涨落和轨道涨落，剩余轨道的电子形成超导配对并负责超导电流的输运。
我们进一步发展了以Lanczos方法为杂质求解器的动力学平均场理论，使其可以计算双轨道Hubbard模型中的超导序参量，并用其研究了双轨道体系中的超导配对。在吸引相互作用下，当两轨道带宽比存在差异时，窄轨道中的超导序参量大于宽轨道，主要是因为此时体系中的电子配对由吸引相互作用所驱动。在特定的相互作用下，带宽差异的体系中存在着仅窄轨道中具有电子配对的轨道选择超导特性，带宽差异能一定程度地促进这种新奇物理现象。轨道间跃迁的引入会使体系中出现不同轨道间的电子配对，同时也会使体系中的轨道选择超导特性消失。不同轨道间的电子配对强度明显小于同轨道内的电子配对，这是由于其配对特征不同于轨道内的电子配对，不同轨道间的电子配对由轨道间跃迁所驱动，而非吸引相互作用。在排斥相互作用下，双轨道体系中超导序参量的数值明显小于吸引相互作用的情况，同时宽轨道中的电子配对强度大于窄轨道，但轨道带宽的增加对超导序参量不存在明显的促进作用，说明排斥相互作用下存在着新的超导配对机制，还有待进一步研究与探索。
文章的结尾我们介绍了之后探索的方向，即进一步发展动力学平均场理论，计算体系中的d-波超导配对序参量以研究铜氧化物材料中的超导机理。最后我们对论文的研究成果进行了总结。

The discovery of cuprate superconductor has brought great challenges to the study of condensed matter theory. Its parent compound is antiferromagnetic Mott insulator, so the strong correlation effect has attracted extensive attention in condensed matter physics.The high degeneracy of $d$-orbitals in transition metallic oxides makes multiorbital effects become an important issue in strongly correlated electronic systems. The high-T_c superconducting(HTSC) compound Ba₂CuO_4−δ discovered in 2019 has a significant two-orbital characteristic, which is different from the traditional HTSC cuprates explained by the Zhang-Rice singlet. So it's necessary to study the quantum phase transition of strong correlated two-orbital system. This paper studies the above problems by using the dynamic mean field theory(DMFT) with Lanczos method as the impurity solver and gets some results which is helpful to research of strong correlated electron system.
After a systematic introduction of the research background and the dynamical mean field theory, we study the orbital-selective Mott transition(OSMT) in the two-orbital Hubbard model. The Hund's rule coupling J promotes the OSMT by blocking orbital fluctuations, and this effect reaches saturation when J ∼ 0.25U. The influence of the interaction U is inconsistent with different bandwidth, and the greater the difference between the two orbital bandwidths, the more likely the OSMT will occur. We introduce inter-orbital hopping in the two-band system, and the decoupled effective orbits can be obtained by the canonical transformation. Inter-orbital hopping promotes the OSMT in the effective two-orbital model by adjusting the hopping integral of effective orbital. Finally, we compared the OSMT in the square lattice and the Bethe lattice. The physical images of different lattice are consistent, but the critical points of phase transition are different. This is mainly due to the difference energy scales. If they are unified, the difference of phase transition will be significantly reduced, so Bethe lattice is widely used in DMFT study.
We extend DMFT with Lanczos solver to handle the bandstructure of Ba₂CuO_4−δ obtained by density functional theory(DFT), and study the influence of electron correlation and hole doping effect on the quantum phase transition of Ba₂CuO_4−δ. We find Ba₂CuO_4−δ with hall-filled e_g orbital display prominent orbital-selective Mott phase(OSMP) character under the correlated effect. Considering Ba₂CuO_4−δ belongs to the transition metallic oxide, we propose a possibility of the parent compound of SC Ba₂CuO_4−δ material, that is, the half-filled orbital selects Mott compound Ba₂CuO_3.5.Under the strong Hund's rule coupling, with the increase of the electron filling of e_g orbital, Ba₂CuO_4−δ system first exhibits OSMP character, then it transforms into the metal phase. According to the density of states(DOS), we find the metallic system should near the edge of OSMP. In Ba₂CuO_3.5 system, in which the SC effect is observed, the electronic states of the two e_g orbitals display different characteristics when the interaction changes, and the system shows obvious two-orbital characteristics. We also find the system has extremely stable orbital polarization in OSMP. Based on our results, we propose a SC mechanism that may exist in the multiband system: the electrons in one or two orbitals can contribute spin or orbital fluctuations, and electrons in another one or two orbitals contribute SC pairs and carry supercurrent.
We further develop the DMFT to calculate the SC order in the two-orbital Hubbard model, and use it to study electron pairings in two-orbital systems. Under the attractive interaction, the narrow orbital SC order is larger than that wide orbital, mainly because the electron pairing is driven by the attractive interaction. In the system with specific interaction and bandwidth difference, there is an orbital-selective superconductivity characteristic in which only narrow orbitals have electron pairings. The inter-orbital hopping will cause electron pairings between different orbitals in the system, and at the same time, the orbital-selective superconductivity in the system will disappear. The strength of electron pairing between different orbitals is significantly smaller than that pairing within the same orbital, which is due to the different pairing mechanism: the electron pairing between different orbitals is driven by inter-orbital transitions rather than attractive interactions. Under the repulsive interaction, the SC order parameter in the two-band system is much smaller than that the case of attractive interaction, and the electron pairing strength in the wide orbital is greater than that in the narrow orbital. The results indicates that there is a new SC pairing mechanism under the repulsive interaction, which needs further research and exploration.
At the end of the paper, we briefly introduce the exploration of our study, that is, to further develop DMFT, and study the d-wave SC pairing in the system so that can learn the SC mechanism in HTSC cuprates. Finally, the research results of this paper are summarized.