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arXiv:1507.06024 · Reviews in Mathematical Physics

The ground state construction of bilayer graphene

Alessandro Giuliani, Ian Jauslin

Published Aug 29, 2016• 2 citations
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Abstract

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.

DOI
10.1142/s0129055x16500185

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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.
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.
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.
Motivated by the recent novel electronic features extracted from the magic-angle graphene superlattices, we studied the ground state superconducting pairing correlations within the Hubbard model on a twisted bilayer honeycomb lattice. Using Constrained-Path Quantum Monte Carlo method, we found that the [Formula: see text] pairing correlation dominates over other pairing patterns among various electron fillings and interaction strengths, and the effective pairing interaction was enhanced as the on-site Coulomb interaction increased. We further examined the effect of the nearest neighbor interaction [Formula: see text], and the effective pairing interaction with [Formula: see text] pairing symmetry was also enhanced by either a repulsive or attractive interaction. Our intensive numerical results confirm the interaction driven superconductivity with a dominant [Formula: see text] pairing symmetry in twisted bilayer graphene.
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.
Monte Carlo simulation of a 2+1 dimensional model of voltage-biased bilayer graphene, consisting of relativistic fermions with chemical potential $\ensuremath{\mu}$ coupled to charged excitations with opposite sign on each layer, has exposed noncanonical scaling of bulk observables near a quantum critical point found at strong coupling. We present a calculation of the quasiparticle dispersion relation E(k)E(k)E(k) as a function of exciton source jjj in the same system, employing partially twisted boundary conditions to boost the number of available momentum modes. The Fermi momentum kF{k}_{F}kF​ and superfluid gap $\mathrm{\ensuremath{\Delta}}$ are extracted in the $j\ensuremath{\rightarrow}0$ limit for three different values of $\ensuremath{\mu}$, and support a strongly interacting scenario at the Fermi surface with $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}O(\ensuremath{\mu})$. We propose an explanation for the observation $\ensuremath{\mu}<{k}_{F}$ in terms of a dynamical critical exponent z<1z<1z<1.
Many-body instabilities of the half-filled honeycomb bilayer are studied using weak-coupling renormalization group (RG) as well as strong-coupling expansion. For spinless fermions and assuming parabolic degeneracy, there are four independent four-fermion contact couplings. While the dominant instability depends on the microscopic values of the couplings, the broken symmetry state is typically a gapped insulator with either broken inversion symmetry or broken time-reversal symmetry, with a quantized anomalous Hall effect. Under certain conditions, the dominant instability may appear in the particle-particle (pairing) channel. For some nongeneric fine-tuned initial conditions, weak-coupling RG trajectories flow into the noninteracting fixed point, although generally we find runaway flows which we associate with ordering tendencies. Additionally, a tight-binding model with nearest-neighbor hopping and nearest-neighbor repulsion is studied in weak and strong couplings and in each regime a gapped phase with inversion symmetry breaking is found. In the strong-coupling limit, the ground-state wave function is constructed for vanishing in-plane hopping but finite interplane hopping, which explicitly displays the broken inversion symmetry and a finite difference between the number of particles on the two layers. Finally, we discuss the spin-1/2 case and use Fierz identities to show that the number of independent four-fermion contact couplings is 9. The corresponding RG equations in the spin-1/2 case are also presented, and used to show that, just as in strong coupling, the most dominant weak-coupling instability of the repulsive Hubbard model (at half filling) is an antiferromagnet.