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arXiv:0901.4867 · Physical Review B

Rigorous construction of ground state correlations in graphene: Renormalization of the velocities and Ward identities

Alessandro Giuliani, Vieri Mastropietro

Published May 20, 2009• 31 citations
arXivHTMLPDF

Abstract

We consider the two-dimensional Hubbard model on the honeycomb lattice, as a model for single-layer graphene with screened Coulomb interactions; at half filling and weak coupling, we construct its ground-state correlations by a convergent multiscale expansion, rigorously excluding the presence of magnetic or superconducting instabilities or the formation of a mass gap. The Fermi velocity, which can be written in terms of a convergent series expansion, remains close to its noninteracting value and turns out to be isotropic; as a consequence, the Dirac cones are isotropic at low energies. On the contrary, the interaction produces an asymmetry between the two components of the charge velocity, in contrast to the predictions based on relativistic or continuum approximations.

DOI
10.1103/physrevb.79.201403

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We consider the two-dimensional Hubbard model on the honeycomb lattice, as a model for single-layer graphene with screened Coulomb interactions; at half filling and weak coupling, we construct its ground-state correlations by a convergent multiscale expansion, rigorously excluding the presence of magnetic or superconducting instabilities or the formation of a mass gap. The Fermi velocity, which can be written in terms of a convergent series expansion, remains close to its noninteracting value and turns out to be isotropic; as a consequence, the Dirac cones are isotropic at low energies. On the contrary, the interaction produces an asymmetry between the two components of the charge velocity, in contrast to the predictions based on relativistic or continuum approximations.
The Hubbard and extended Hubbard models on the honeycomb lattice can be seen as prototype models of a single-layer graphene placed in a high dielectric constant environment that screens the Coulomb interaction. Taking advantage of the absence of a sign problem at half-filling, we study this problem with clusters up to 96 sites with the determinant quantum Monte Carlo method as an impurity solver for the dynamical cluster approximation at finite temperatures. After determining the stability of the semimetallic phase to interaction-induced spin-density wave (SDW), charge-density wave (CDW), and Mott insulating phases, we study the single-particle dynamics of the Dirac fermions. We show that when spontaneous symmetry breaking is avoided, the semimetallic phase is a stable Fermi liquid in the presence of repulsive interactions and that Kondo screening dominates the low temperature regime, even though there is a $\ensuremath{\rho}(\ensuremath{\omega})=|\ensuremath{\omega}|$ type local density of states. We also investigate the impact of the correlation effects on the renormalization of the Fermi velocity vF{v}_{F}vF​. We find that vF{v}_{F}vF​ is not renormalized when only on-site repulsion UUU is present, but that near-neighbor repulsion VVV does renormalize vF{v}_{F}vF​. This may explain the variations between different measurements of vF{v}_{F}vF​ in graphene.
We consider a model of half-filled bilayer graphene, in which the three dominant Slonczewski–Weiss–McClure hopping parameters are retained, in the presence of short-range interactions. Under a smallness assumption on the interaction strength [Formula: see text] as well as on the inter-layer hopping [Formula: see text], we construct the ground state in the thermodynamic limit, and prove that the pressure and two-point Schwinger function, away from its singularities, are analytic in [Formula: see text], uniformly in [Formula: see text]. The interacting Fermi surface is degenerate, and consists of eight Fermi points, two of which are protected by symmetries, while the locations of the other six are renormalized by the interaction, and the effective dispersion relation at the Fermi points is conical. The construction reveals the presence of different energy regimes, where the effective behavior of correlation functions changes qualitatively. The analysis of the crossover between regimes plays an important role in the proof of analyticity and in the uniform control of the radius of convergence. The proof is based on a rigorous implementation of fermionic renormalization group methods, including determinant estimates for the renormalized expansion.
Both insulating and conducting electronic behaviors have been experimentally seen in clean bilayer graphene samples at low temperature, and there is still no consensus on the nature of the interacting ground state at half filling and in the absence of a magnetic field. Theoretically, several possibilities for the insulating ground states have been predicted for weak interaction strength. However, a recent renormalization-group calculation on a Hubbard model for charge-neutral bilayer graphene with short-range interactions suggests the emergence of low-energy Dirac fermions that would stabilize the metallic phase for weak interactions. Using a nonperturbative projective quantum Monte Carlo, we calculate the ground state for bilayer graphene using a realistic model for the Coulomb interaction that includes both short-range and long-range contributions. We find that a finite critical on-site interaction is needed to gap bilayer graphene and the transition belongs to the Gross-Neveu universality class, thereby confirming the Hubbard model expectations even in the presence of a long-range Coulomb potential, in agreement with our theoretical analysis. In addition, we also find that the critical on-site interactions necessary to destabilize the metallic ground state decrease with increasing interlayer coupling.
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.