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Development in Variational Gutzwiller Theory: From Formalism to Applications in Twisted Bilayer Graphene

Shu Liang

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This thesis develops variational Gutzwiller theory for realistic multi-orbital strongly correlated systems and applies it to twisted bilayer graphene (TBG), where correlated insulating, metallic, and superconducting phases appear in close proximity. We first establish an efficient and robust implementation of the multi-orbital Gutzwiller method, including the original-basis formulation, stable variational equations, analytical Jacobians and gradients, practical density-matrix parametrizations, and the use of gauge freedom to reduce computational cost. Benchmark calculations demonstrate that this implementation captures correlation effects beyond Hartree-band model of TBG with correlated 𝑓orbitals and itinerant 𝑐orbitals. The resulting phase diagram reveals a Fermi-liquid dome that separates a weakly correlated BCS-like superconducting regime from a strongly correlated superconducting regime. In the latter, pairing is tied to local spin-valley-orbital structure within a moiré unit cell and is better understood in terms of preformed local pairs that gain itinerancy through charge fluctuations and hybridization. Finally, we develop a linear-response framework within the variational Gutzwiller method and derive an analytical expression for the static compressibility. Applied to symmetric TBG, this formalism naturally explains the cascade-like compressibility minima as a consequence of strong correlations in the localized orbitals. Together, these results establish variational Gutzwiller theory as a powerful framework for studying correlated normal and superconducting phases in moirématerials.

DOI
10.14711/thesis-hdl174672

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This thesis develops variational Gutzwiller theory for realistic multi-orbital strongly correlated systems and applies it to twisted bilayer graphene (TBG), where correlated insulating, metallic, and superconducting phases appear in close proximity. We first establish an efficient and robust implementation of the multi-orbital Gutzwiller method, including the original-basis formulation, stable variational equations, analytical Jacobians and gradients, practical density-matrix parametrizations, and the use of gauge freedom to reduce computational cost. Benchmark calculations demonstrate that this implementation captures correlation effects beyond Hartree-band model of TBG with correlated 𝑓orbitals and itinerant 𝑐orbitals. The resulting phase diagram reveals a Fermi-liquid dome that separates a weakly correlated BCS-like superconducting regime from a strongly correlated superconducting regime. In the latter, pairing is tied to local spin-valley-orbital structure within a moiré unit cell and is better understood in terms of preformed local pairs that gain itinerancy through charge fluctuations and hybridization. Finally, we develop a linear-response framework within the variational Gutzwiller method and derive an analytical expression for the static compressibility. Applied to symmetric TBG, this formalism naturally explains the cascade-like compressibility minima as a consequence of strong correlations in the localized orbitals. Together, these results establish variational Gutzwiller theory as a powerful framework for studying correlated normal and superconducting phases in moirématerials.
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.
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.
The rich phenomenology of twisted bilayer graphene (TBG) near the magic angle is believed to arise from electron correlations in topological flat bands.An unbiased approach to this problem is highly desirable, but also particularly challenging, given the multiple electron flavors, the topological obstruction to defining tight-binding models, and the long-ranged Coulomb interactions.While numerical simulations of realistic models have thus far been confined to zero temperature, typically excluding some spin or valley species, analytic progress has relied on fixed point models away from the realistic limit.Here, we present unbiased Monte Carlo simulations of realistic models of magic-angle TBG at charge neutrality.We establish the absence of a sign problem for this model in a momentumspace approach and describe a computationally tractable formulation that applies even on breaking chiral symmetry and including band dispersion.Our results include (i) the emergence of an insulating Kramers intervalley coherent ground state in competition with a correlated semimetal phase, (ii) detailed temperature evolution of order parameters and electronic spectral functions that reveal a "pseudogap" regime, in which gap features are established at a higher temperature than the onset of order, and (iii) predictions for electronic tunneling spectra and their evolution with temperature.Our results pave the way towards uncovering the physics of magic-angle graphene through exact simulations of over a hundred electrons across a wide temperature range.
We use stochastic expansion and exact diagonalization to study magic-angle twisted bilayer graphene (TBG) on a disordered substrate. We show that the substrate-induced strong Coulomb disorder in TBG with the chemical potential at the level of the flat bands drives the system to a network of weakly coupled Sachdev-Ye-Kitaev (SYK) bundles, stabilizing an emergent quantum chaotic strange metal (SM) phase of TBG that exhibits the absence of quasiparticles. The Gaussian orthogonal ensemble dominates TBG's long-time chaotic dynamics at strong disorder, whereas fast quantum scrambling appears in the short-time dynamics. In weak disorder, gapped phases of TBG exhibit exponentially decaying specific heat capacity and exponential decay in out-of-time-ordered correlators (OTOCs). This is the system behavior in correlated insulator and superconducting phases, in agreement with the corresponding Larkin-Ovchinnikov result for correlators. The result suggests a low-temperature transition from the superconducting and correlated insulating phases into the strange metal upon increasing the disorder strength. We propose a finite-temperature phase diagram for Coulomb-disordered TBG and discuss the experimental consequences of the emergent SM phase.
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.
Understanding the material parameters that control the superconducting transition temperature T c is a problem of fundamental importance.In many novel superconductors phase fluctuations determine T c , rather than the collapse of the pairing amplitude.We derive rigorous upper bounds on the superfluid phase stiffness for multiband systems, valid in any dimension.This in turn leads to an upper bound on T c in two dimensions, which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry.Our bound is particularly useful for the strongly correlated regime of low-density and narrowband systems, where mean-field theory fails.For a simple parabolic band in 2D with Fermi energy E F , we find that k B T c ≤ E F =8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases.Applying our multiband bound to magic-angle twisted bilayer graphene, we find that band structure results constrain the maximum T c to be close to the experimentally observed value.Finally, we discuss the question of deriving rigorous upper bounds on T c in 3D.