图 1  (a)扭角双层石墨烯有效两轨道哈伯德模型电子跳跃项示意图. 黑色(红色)点代表子格子A (B), 每个晶格点包含两个轨道. $ t_{1}, t_{1}' $表示最近邻电子跳跃积分, $ t_{2}, t_{2}' $代表第五近邻电子跳跃积分. (b)扭转角度为1.08°、晶格尺寸$ L = 5 $的晶格结构示意图. 黑色和红色子格点表示第一层不等价的碳原子A和碳原子B, 绿色和蓝色表示第二层不等价的碳原子$ A_{1} $和碳原子$ B_{1} $. 三条黑色的曲线表示周期性结构边长

Figure 1.  (a) Schematic diagram illustrating the electron hopping terms of effective two-orbital Hubbard model on the twisted bilayer graphene. The black (red) dots represent sublattice A (B), with each lattice point containing two orbitals. The hopping integrals $ t_{1} $ and $ t_{1}' $ correspond to nearest-neighbor interactions, while $ t_{2} $ and $ t_{2}' $ represent the fifth-nearest-neighbor interactions. (b) Schematic of the lattice structure with a twist angle of 1.08° and lattice size $ L = 5 $. Black and red points correspond to the inequivalent carbon atoms A and B in the first layer, while green and blue points represent the inequivalent atoms $ A_{1} $ and $ B_{1} $ in the second layer. The three black curves denote the periodic boundary lengths of the structure.

图 2  各种轨道内电子配对形式的示意图　(a) NN-s对称性; (b) NN-$ {\mathrm{d}}+{\mathrm{id}} $对称性; (c) NN-$ {\mathrm{p}}+{\mathrm{ip}} $对称性; (d) NNN-$ {\mathrm{d}}+{\mathrm{id}} $对称性; (e) NNN-$ {\mathrm{p}}+{\mathrm{ip}} $对称性; (f) NNN-f对称性

Figure 2.  Schematic diagrams of various intra-orbital electron pairing symmetry: (a) NN-s-wave symmetry; (b) NN-$ {\mathrm{d}}+{\mathrm{id}} $-wave symmetry; (c) NN-$ {\mathrm{p}}+{\mathrm{ip}} $-wave symmetry; (d) NNN-$ {\mathrm{d}}+{\mathrm{id}} $-wave symmetry; (e) NNN-${\mathrm{p}}+{\mathrm{ip}} $-wave symmetry; (f) NNN-f-wave symmetry.

图 3  晶格大小为$ L = 5 $的各种电子配对对称性的配对关联函数$ P_{\alpha}(R) $随配对距离R的变化　(a) $ U = 2.0,V =0.0, $$ \langle n \rangle = 0.933 $; (b) $ U = 0.0, V = -0.3, \langle n \rangle = 0.933 $; (c) $ U = 2.0, V = 0.0, \langle n \rangle = 0.893 $; (d) $ U = 0.0,V = -0.3,\langle n \rangle =0.893 $

Figure 3.  Pairing correlation functions $ P_{\alpha}(R) $ as a function of pairing distance R for various electron pairing symmetries in a lattice of size $ L = 5 $: (a) $ U = 2.0 $, $ V = 0.0 $, $ \langle n \rangle = 0.933 $; (b) $ U = 0.0 $, $ V = -0.3 $, $ \langle n \rangle = 0.933 $; (c) $ U = 2.0 $, $ V = 0.0 $, $ \langle n \rangle = 0.893 $; (d) $ U = 0.0 $, $ V = -0.3 $, $ \langle n \rangle = 0.893 $.

图 4  无相互作用项时晶格大小为$ L = 5 $的各种电子配对对称性的配对关联函数$ P_{\alpha}(R) $随配对距离R的变化　(a) $ \langle n \rangle = $$ 0.933 $; (b) $ \langle n \rangle = 0.893 $

Figure 4.  Pairing correlation functions $ P_{\alpha}(R) $ as a function of pairing distance R for various electron pairing symmetries in a non-interacting system with lattice size $ L = 5 $: (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $.

图 5  在位库仑相互作用强度$ U = 2.0 $时, 晶格大小为$ L = 5 $的各种电子配对对称性的平均有效配对关联函数$ \overline{V}_{\alpha}(R\geqslant3) $随近邻库仑相互作用强度V的变化　(a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $

Figure 5.  Average effective pairing correlation functions $ \overline{V}_{\alpha}(R\geqslant3) $ as a function of nearest-neighbor Coulomb interaction V for various electron pairing symmetries in a system with on-site Coulomb interaction strength $ U = 2.0 $ and lattice size $ L = 5 $: (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $.

图 6  在位库仑相互作用强度$ U = 2.0 $时, 晶格大小为$ L = 6 $的手性NN-$ {\mathrm{d}}+{\mathrm{id}} $波配对对称性的有效配对关联函数$ {V}_{{\mathrm{d}}+{\mathrm{id}}} $随长程配对距离R的变化　(a) $ \langle n \rangle = 0.954 $; (b) $ \langle n \rangle = 0.926 $

Figure 6.  Effective pairing correlation functions $ {V}_{{\mathrm{d}}+{\mathrm{id}}} $ as a function of long-range pairing distance R for the chiral NN-$ {\mathrm{d}}+{\mathrm{id}} $-wave pairing symmetry in a system with on-site Coulomb interaction strength $ U = 2.0 $ and lattice size $ L = 6 $: (a) $ \langle n \rangle = 0.954 $; (b) $ \langle n \rangle = 0.926 $.

图 7  在位库仑相互作用强度$ U = 2.0 $时, 晶格大小为$ L = 5 $的自旋结构因子$ S(q) $沿着第一布里渊区高对称线$ \varGamma \to M \to $$ K \to \varGamma $方向的变化曲线　(a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $. 插图中的紫色线代表第一布里渊区的高对称线, 这里Γ, M, K的坐标分别为$ (0, 0) $, $ \Big(\dfrac{2\pi}{3}, 0\Big) $, $ \Big(\dfrac{2\pi}{3\sqrt{3}}, 0\Big) $

Figure 7.  Spin structure factor $ S(q) $ along the high-symmetry lines $ \varGamma \to M \to K \to \varGamma $ in the first Brillouin zone for a system with on-site Coulomb interaction strength $ U = 2.0 $ and lattice size $ L = 5 $: (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $. The inset shows the high-symmetry lines in the first Brillouin zone, with the coordinates of Γ, M, and K given by $ (0, 0) $, $\Big(\dfrac{2\pi}{3}, 0\Big) $, and $ \Big(\dfrac{2\pi}{3\sqrt{3}}, 0\Big) $, respectively. The purple lines in the inset represent the high-symmetry lines.

图 8  无相互作用的哈伯德模型对应的能带结构和态密度　(a), (c), (e)分别为$ t_{1}' = t_{2}' = 0.15 $, $ t_{1}' = t_{2}' = 0.10 $, $ t_{1}' = t_{2}' = 0.05 $的情况下能带沿着第一布里渊区高对称线方向的演化曲线; (b), (d), (f)分别为$ t_{1}' = t_{2}' = 0.15 $, $ t_{1}' = t_{2}' = 0.10 $, $ t_{1}' = t_{2}' = 0.05 $的情况下状态数关于能量的函数关系, 红色虚线和蓝色虚线分别对应电子填充浓度为$ \langle n \rangle = 0.933 $和$ \langle n \rangle = 0.893 $的费米能级位置

Figure 8.  Band structure and density of states (DOS) for the non-interacting Hubbard model. Panels (a), (c), and (e) display the band dispersion along the high-symmetry lines in the first Brillouin zone for $ t_{1}' = t_{2}' = 0.15 $, $ t_{1}' = t_{2}' = 0.10 $, and $ t_{1}' = t_{2}' = 0.05 $, respectively. Panels (b), (d), and (f) show the density of states as a function of energy for the same values of $ t_{1}' = t_{2}' $. The red and blue dashed lines represent the Fermi level positions corresponding to electron fillings of $ \langle n \rangle = 0.933 $ and $ \langle n \rangle = 0.893 $, respectively.

图 9  在位库仑相互作用强度$ U = 2.0 $及近邻库仑相互作用强度$ V = 0.0 $时, 晶格大小为$ L = 5 $的各种电子配对对称性的平均有效配对关联函数$ \overline{V}_{\alpha}(R\geqslant3) $随电子跳跃破缺项$ t_{1, 2}' $的变化　(a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $

Figure 9.  Average effective pairing correlation functions $ \overline{V}{\alpha}(R \geqslant 3) $ as a function of electron hopping anisotropy terms $ t_{1, 2}' $ for various electron pairing symmetries in a system with on-site Coulomb interaction strength $ U = 2.0 $, nearest-neighbor Coulomb interaction strength $ V = 0.0 $, and lattice size $ L = 5 $: (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $.

图 10  平带结构调控下, 在位库仑相互作用强度$ U = 2.0 $及近邻库仑相互作用强度$ V = 0.0 $时, 晶格大小为$ L = 6 $的手性NN-$ {\mathrm{d}}+{\mathrm{id}} $波配对对称性有效配对关联函数$ V_{{\mathrm{d}}+{\mathrm{id}}} $随长程配对距离R的变化　(a) $ \langle n \rangle = 0.954 $; (b) $ \langle n \rangle = 0.926 $

Figure 10.  Effective pairing correlation function $ V_{{\mathrm{d}}+{\mathrm{id}}} $ as a function of the long-range pairing distance R for the chiral NN-$ {\rm d+id} $-wave pairing symmetry with lattice size $ L = 6 $ under flat band structure modulation with an on-site Coulomb interaction strength of $ U = 2.0 $ and nearest-neighbor Coulomb interaction strength of $ V = 0.0 $: (a) $ \langle n \rangle = 0.954 $; (b) $ \langle n \rangle = 0.926 $.

图 11  平带结构调控下, 在位库仑相互作用强度$ U = 2.0 $、近邻库仑相互作用强度$ V = 0.0 $、晶格大小为$ L = 5 $时, 自旋结构因子$ S(q) $沿着第一布里渊区高对称线$ \varGamma \to M \to K \to \varGamma $方向的变化曲线　(a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $

Figure 11.  Spin structure factor $ S(q) $ along the high-symmetry lines $ \varGamma \to M \to K \to \varGamma $ in the first Brillouin zone for a system with on-site Coulomb interaction strength $ U = 2.0 $, nearest-neighbor Coulomb interaction strength $ V = 0.0 $, and lattice size $ L = 5 $, under flat band structure modulation: (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $.