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The Journal of Physical Chemistry C

Electron Correlations Study toward Understanding Twisted Bilayer Graphene: Two-Orbital Hubbard Model and Exact Numerical Calculations

Florentino Lopéz‐Urías, Francisco Sánchez-Ochoa

Published Feb 9, 2026• 0 citations
Publisher

Abstract

We investigated the honeycomb lattice using a two-orbital (α and β) Hubbard model and many-body exact calculations. The spin gap (Δ s ), charge gap (Δ c ), intrasite ( γ i i α β = ⟨ S → i α · S → i β ⟩ ), intersite ( γ i j α β = ⟨ S → i α · S → j β ⟩ ), spin–spin correlations, and orbital magnetic moments μ iα are analyzed for different band-filling ( n ), which denotes the number of electrons per site. Special attention was set in one-quarter-filled ( n = 1) and half-filled ( n = 2) bands. The results are presented by varying Coulomb repulsion ( U), Hund’s coupling ( J H ), external magnetic field ( B ), and orbital energy splitting (Δϵ). For the one-quarter-filled band, an intrasite magnetic disorder ( γ i i α β ≈ 0 ) begins to form for large values of U / t, which is interpreted as a possible signature of a resonating valence bond state. Full orbital polarization ( γ i i α β = 0.25 ) with intersite antiferromagnetic (AFM) order ( γ i j α β = − 0.25 ) was observed in the half-filled band for U / t ≫ 1, indicating that the two-orbital Hubbard model can be mapped to an AFM Heisenberg model with S = 1. The increase of J H is conductive to insulating and metallic phases for half- and one-quarter-filled bands, respectively. For a fractional n, the honeycomb lattice can exhibit spin-glass-like behavior, predominating as a disordered magnetic system characterized by random values of γ i j α β . The magnetic field induced a reduction in Δ c, which became zero at the critical external magnetic field. Twisted bilayer graphene (TBG) exhibits approximately flat electronic bands at magic angles and interesting correlated phenomena, such as unconventional superconductivity and correlated insulating states around n = 1 and n = 2, respectively. The dominance of electron–electron interactions in TBG makes the Hubbard model, in various versions (extended or multi-orbital), offer a practical, structure-scale account for investigating many-body effects in large moiré unit cells, where first-principles density-functional theory calculations are computationally challenging.

DOI
10.1021/acs.jpcc.5c05791

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We investigated the honeycomb lattice using a two-orbital (α and β) Hubbard model and many-body exact calculations. The spin gap (Δ s ), charge gap (Δ c ), intrasite ( γ i i α β = ⟨ S → i α · S → i β ⟩ ), intersite ( γ i j α β = ⟨ S → i α · S → j β ⟩ ), spin–spin correlations, and orbital magnetic moments μ iα are analyzed for different band-filling ( n ), which denotes the number of electrons per site. Special attention was set in one-quarter-filled ( n = 1) and half-filled ( n = 2) bands. The results are presented by varying Coulomb repulsion ( U), Hund’s coupling ( J H ), external magnetic field ( B ), and orbital energy splitting (Δϵ). For the one-quarter-filled band, an intrasite magnetic disorder ( γ i i α β ≈ 0 ) begins to form for large values of U / t, which is interpreted as a possible signature of a resonating valence bond state. Full orbital polarization ( γ i i α β = 0.25 ) with intersite antiferromagnetic (AFM) order ( γ i j α β = − 0.25 ) was observed in the half-filled band for U / t ≫ 1, indicating that the two-orbital Hubbard model can be mapped to an AFM Heisenberg model with S = 1. The increase of J H is conductive to insulating and metallic phases for half- and one-quarter-filled bands, respectively. For a fractional n, the honeycomb lattice can exhibit spin-glass-like behavior, predominating as a disordered magnetic system characterized by random values of γ i j α β . The magnetic field induced a reduction in Δ c, which became zero at the critical external magnetic field. Twisted bilayer graphene (TBG) exhibits approximately flat electronic bands at magic angles and interesting correlated phenomena, such as unconventional superconductivity and correlated insulating states around n = 1 and n = 2, respectively. The dominance of electron–electron interactions in TBG makes the Hubbard model, in various versions (extended or multi-orbital), offer a practical, structure-scale account for investigating many-body effects in large moiré unit cells, where first-principles density-functional theory calculations are computationally challenging.
Motivated by recent experimental studies that have found signatures of a correlated insulator phase and tuning superconductivity in twisted bilayer graphene, we study the temperature-dependent conductivity, the spin correlation and the superconducting pairing correlation within a two-orbital Hubbard model on an emergent honeycomb lattice. Evaluation of the temperature dependence of the conductivity demonstrates that there is a metal-insulator transition, and the Mott phase at strong coupling is accompanied by antiferromagnetic order. The electronic correlation drives a d+idd+idd+id superconducting pairing to be dominant over a wide filling region. All of the dc conductivity, the spin correlation and the superconductivity are suppressed as the interlayer coupling strength increases, and the critical UcU_cUc​ for metal-insulator transition is also reduced. Our intensive numerical results reveal that the TBG should be a uniquely tunable platform for exploring correlated states.
Motivated by recent experimental studies that have found signatures of a correlated insulator phase and tuning superconductivity in twisted bilayer graphene, we study the temperature-dependent conductivity, the spin correlation, and the superconducting pairing correlation within a two-orbital Hubbard model on an emergent honeycomb lattice. The evaluation of the temperature dependence of the conductivity demonstrates that there is a metal-insulator transition and the Mott phase at strong coupling is accompanied by antiferromagnetic order. The electronic correlation drives a d+idd+idd+id superconducting pairing to be dominant over a wide filling region. All of the dc conductivity, the spin correlation, and the superconductivity are suppressed as the interlayer coupling strength increases, and the critical Uc{U}_{c}Uc​ for the metal-insulator transition is also reduced. Our intensive numerical results reveal that twisted bilayer graphene should be a uniquely tunable platform for exploring strongly correlated phenomena.
Resonating valence bond (RVB) states are fundamental for understanding quantum spin liquids in two-dimensional (2D) systems. The RVB state is a collective phenomenon in which spins are uncoupled. 2D lattices such as triangular, honeycomb, and dice lattices were investigated using the Hubbard model and exact diagonalization method. We analyzed the total spin, spin-spin correlation functions, local magnetic moments, and spin and charge gaps as a function of on-site Coulomb repulsion, electron concentration, and electronic hopping parameters. Phase diagrams showed that RVB states can live in half-filled and hole-doped anisotropic triangular lattices. We found two types of RVB states: one in the honeycomb sublattice and the other in the centered hexagons in the triangular lattices. Owing to the novel discovery of exotic magnetic ordering in triangular moiré patterns in twisted bilayer graphene and transition metal dichalcogenide systems, our results provide physical insights into the onset of magnetism and possible spin liquid states in these layered materials.
To understand correlated insulating and unconventional superconducting states in twisted bilayer graphene, we perform a systematic study of spin and pairing correlations in an effective two-orbital Hubbard model, by using the ground-state constrained-path quantum Monte Carlo method. Our numerical simulations reveal that when only the on-site Hubbard UUU is considered, the long-range magnetic order develops for $U\ensuremath{\ge}3.2t$ at half filling. Upon doping away from half filling, an analysis based on the pairing correlations and corresponding vertex contributions identifies that pairing with the d+idd+idd+id symmetry is the dominant pairing channel. As the on-site Coulomb interaction is increased, both spin correlation and effective pairing interaction are enhanced simultaneously, indicating the intimate relation between magnetism and superconductivity. An inclusion of nearest neighbor Coulomb interaction VVV results in a suppression of the d+idd+idd+id superconductivity, but it can survive in the small VVV regime. Our findings are useful for clarifying the ongoing controversy on the superconducting pairing symmetry in twisted bilayer graphene.
We study the possibility of charge order at quarter filling and antiferromagnetism at half-filling in a tight-binding model of magic angle twisted bilayer graphene. We build on the model proposed by Kang and Vafek [1], relevant to a twist angle of 1.30^\circ 1.30 ∘ , and add on-site and extended density-density interactions. Applying the variational cluster approximation with an exact-diagonalization impurity solver, we find that the system is indeed a correlated (Mott) insulator at fillings \frac14 1 4 , \frac12 1 2 and \frac34 3 4 . At quarter filling, we check that the most probable charge orders do not arise, for all values of the interaction tested. At half-filling, antiferromagnetism only arises if the local repulsion U U is sufficiently large compared to the extended interactions, beyond what is expected from the simplest model of extended interactions.
Magic-angle twisted bilayer graphene has recently become a thriving material platform realizing correlated electron phenomena taking place within its topological flat bands. Several numerical and analytical methods have been applied to understand the correlated phases therein, revealing some similarity with the quantum Hall physics. In this work, we provide a Mott-Hubbard perspective for the TBG system. Employing the large-scale density matrix renormalization group on the lattice model containing the projected Coulomb interactions only, we identify a first-order quantum phase transition between the insulating stripe phase and the quantum anomalous Hall state with the Chern number of ±1. Our results not only shed light on the mechanism of the quantum anomalous Hall state discovered at three-quarters filling, but also provide an example of the topological Mott insulator, i.e., the quantum anomalous Hall state in the strong coupling limit.