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We employ a large-scale, unbiased constrained-path quantum Monte Carlo method to systematically simulate the effective two-orbital Hubbard model for twisted bilayer graphene in order to gain deeper insight into the relationship between correlated states and the superconducting pairing mechanism in twisted bilayer graphene, as well as the influence of the twist angle on superconductivity. Initially, we investigate the modulation of superconductivity by nearest-neighbor attractive Coulomb interactions, demonstrating that electron-phonon coupling plays a significant role in the system. Our numerical results reveal that the superconducting state is dominated by chiral NN-${\mathrm{d}}+{\mathrm{id}} $ superconducting electron pairing symmetry, and that such nearest-neighbor attractive Coulomb interactions significantly enhance the effective long-range pairing correlation function of chiral NN-${\mathrm{d}}+{\mathrm{id}} $ wave. From this perspective, it is evident that the electron-phonon coupling positively contributes to the superconductivity of the system. Then, we explore how the twist angle affects the superconducting state. The flat-band structure caused by hopping anisotropy reflects the different twist angles of the system. Our results show that as the twist angle deviates downward from 1.08°, the effective pairing correlation function of the chiral NN-${\mathrm{d}}+{\mathrm{id}} $ wave increases substantially. Conversely, as the twist angle exceeds 1.08°, the effective correlation function of the chiral NN-${\mathrm{d}}+{\mathrm{id}} $ wave exhibits a tendency of decline. These results suggest that further reduction of the twist angle may lead to higher superconducting transition temperature in twisted bilayer graphene system. Finally, we analyze how nearest-neighbor attractive Coulomb interactions and flat-band structures influence superconductivity from the standpoint of magnetic properties. The observed enhancement of the spin structure factor near the Γ point in the Brillouin zone indicates that enhanced antiferromagnetic correlations are essential for enhancing the superconducting transition temperature and for stabilizing chiral NN-${\mathrm{d}}+{\mathrm{id}} $ wave. Through these investigations, our numerical findings not only contribute to a more comprehensive understanding of strongly correlated systems such as twisted bilayer graphene, but also provide guidance for identifying twist-angle systems with potentially higher superconducting transition temperatures. -
Keywords:
- chiral $\mathrm{d}+\mathrm{id} $ superconducting state /
- constrained path quantum Monte Carlo method /
- twisted bilayer graphene /
- magnetism
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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}' $代表第五近邻电子跳跃积分; (b) 扭转角度为1.08°, 晶格尺寸为$ L = 5 $的晶格结构示意图. 黑色和红色子格点表示第一层不等价的碳原子A和碳原子B, 绿色和蓝色表示第二次层不等价的碳原子$ 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 to Hubbard obital 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) $关于配对距离R的函数曲线 (a) $ U = 2.0,V =0.0, $$ \langle n \rangle = 0.933 $; (b) $ U = 0.0, V = -0.3, \langle n \rangle = 0.933 $; (c) $ U = 2.0, V = 0.0, \langle n \rangle = 0.893 $; (d) $ U = 0.0,V = -0.3,\langle n \rangle =0.893 $
Figure 3. Pairing correlation functions $ P_{\alpha}(R) $ as a function of pairing distance R for various electron pairing symmetries in a lattice of size $ L = 5 $ (a) $ U = 2.0 $, $ V = 0.0 $, $ \langle n \rangle = 0.933 $; (b) $ U = 0.0 $, $ V = -0.3 $, $ \langle n \rangle = 0.933 $; (c) $ U = 2.0 $, $ V = 0.0 $, $ \langle n \rangle = 0.893 $; (d) $ U = 0.0 $, $ V = -0.3 $, $ \langle n \rangle = 0.893 $.
图 4 无相互作用项时晶格大小为$ L = 5 $的各种电子配对对称性的配对关联函数$ P_{\alpha}(R) $关于配对距离R的函数曲线 (a) $ \langle n \rangle = $$ 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) $关于近邻库仑相互作用强度V的函数曲线 (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}}} $关于长程配对距离R的函数关系 (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) $沿着第一布里渊区高对称线$ \Gamma \to M \to $$ K \to \Gamma $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) $, $ (\dfrac{2\pi}{3}, 0) $, $ (\dfrac{2\pi}{3\sqrt{3}}, 0) $
Figure 7. Spin structure factor $ S(q) $ along the high-symmetry lines $ \Gamma \to M \to K \to \Gamma $ 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) $, $ (\dfrac{2\pi}{3}, 0) $, and $ (\dfrac{2\pi}{3\sqrt{3}}, 0) $, respectively. The purple lines in the inset represent the high-symmetry lines.
图 8 无相互作用的哈伯德模型对应的能带结构和态密度; (a), (c), (e)分别为$ 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. The band structure and density of states (DOS) for the non-interacting Hubbard model are shown as follows: 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) $关于电子跳跃破缺项$ t_{1, 2}' $的函数曲线 (a) $ \langle n \rangle = 0.933 $; (b) $ \langle n \rangle = 0.893 $
Figure 9. The 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}}} $关于长程配对距离R的函数曲线 (a) $ \langle n \rangle = 0.954 $; (b) $ \langle n \rangle = 0.926 $
Figure 10. The effective pairing correlation function $ V_{{\mathrm{d}}+{\mathrm{id}}} $ for the chiral NN-$ d+id $-wave pairing symmetry with lattice size $ L = 6 $, under flat band structure modulation, is plotted as a function of the long-range pairing distance R, 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 $.
图 11 在位库仑相互作用强度$ 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 $的平带结构调控下的自旋结构因子$ S(q) $沿着第一布里渊区高对称线$ \Gamma \to M \to K \to \Gamma $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 $ \Gamma \to M \to K \to \Gamma $ 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 $.
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