Electronic ground state in bilayer graphene with realistic Coulomb interactions

Jia Ning Leaw, Ho-Kin Tang, Pinaki Sengupta, Fakher F. Assaad, Igor F. Herbut, and Shaffique Adam

Phys. Rev. B 100, 125116 – Published 6 September 2019

Abstract

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.

Received 15 March 2019
Revised 16 August 2019

DOI:https://doi.org/10.1103/PhysRevB.100.125116

Physics Subject Headings (PhySH)

Electronic structure

Graphene

Hubbard model Quantum Monte Carlo

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jia Ning Leaw and Ho-Kin Tang

Department of Physics and Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546 Singapore, Singapore

Pinaki Sengupta

Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546 Singapore, Singapore and School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore, Singapore

Fakher F. Assaad

Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

Igor F. Herbut

Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

Shaffique Adam

Department of Physics and Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546 Singapore, Singapore and Yale-NUS College, 16 College Avenue West, 138614 Singapore, Singapore

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Issue

Vol. 100, Iss. 12 — 15 September 2019

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Images

Figure 1
Finite size scaling analysis for Hubbard $(γ = 0)$ and Coulomb $(γ = 1)$ interactions with $t_{⊥} = 5 t, 10 t$ . In the main panels, the correlation ratios $R$ for different system sizes are plotted against the on-site interaction strength. The crossings of different data sets show the scale-invariance feature, indicating the phase transition. The interaction strengths at which the crossings occur are identified as the critical interactions $U_{c}$ , with their estimated values listed on top in units of intralayer hopping $t$ . In the insets, the data sets collapse to a single curve when they are scaled with the critical exponents $ν = 0.88$ , showing that the phase transition belongs to the appropriate Gross-Neveu universality class [37, 38, 39, 40, 41].
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Figure 2
The critical on-site interaction strength $U_{c}$ for systems with different interlayer hopping $t_{⊥}$ . (a) The phase boundary for different $t_{⊥}$ in the phase space of on-site interaction $U$ and long-range interaction $α$ . To the left of the phase boundaries is the semimetallic phase, while to the right of the phase boundaries is the antiferromagnetic Mott insulating phase. The critical on-site interaction strength $U_{c}$ decreases when we increase the interlayer hopping $t_{⊥}$ from $t_{⊥} = 0$ , which is the case of monolayer graphene, to $t_{⊥} = 10 t$ . The critical on-site interaction strength $U_{c}$ remains finite and increases when the long-range interaction $α$ is turned on. The shaded window shows the region inaccessible to our quantum Monte Carlo method. (b) The critical on-site interaction strength $U_{c}$ is plotted against the interlayer hopping $t_{⊥}$ for the Hubbard model.
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Figure 3
The noninteracting spectrum for $t_{⊥} = t$ and $10 t$ . The data points show the noninteracting energies at the discrete $k$ points accessible by the largest system size $L = 15$ considered in our quantum Monte Carlo simulations. The black curves are the linear and quadratic fits to the points under the curves. For interlayer coupling $t_{⊥} = t$ , although the spectrum is parabolic in the long-wavelength limit, in the systems with finite size and thus discretized $k$ points, the linearity of the noninteracting spectrum is still discernible. For large interlayer coupling $t_{⊥} = 10 t$ , the noninteracting spectrum becomes parabolic.
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Figure 4
The energy renormalization for interacting electrons in bilayer graphene. The first row shows the energy renormalization $(E - E_{0}) / t$ of the systems with large interlayer hopping $t_{⊥} = 10 t$ , where the noninteracting band is expected to be parabolic. In both cases of purely on-site interaction $γ = 0$ (top-left panel) and with the long-range interaction $γ = 1$ (top-right panel), the energy renormalizes to a larger value. The second row shows the energy renormalization of the systems with $t_{⊥} = t$ , which share similar features with monolayer graphene. In monolayer graphene, there is no energy renormalization in the first order of the Hubbard interaction, but the dispersion is suppressed when we tune up further the Hubbard interaction; in the presence of long-range interaction, the energy renormalization is greatly enhanced due to the Fermi velocity divergence (see Ref. [24]). We see a weak suppression in the spectrum for systems with Hubbard interaction (bottom-left panel), and large energy renormalization with the inclusion of long-range interaction (bottom-right panel). The offset at the band-touching points is due to the finite-size effect.
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Figure 5
The $q$ dependence of the linear term in $α$ (see text). Since this term is nonzero for any finite $t_{⊥}$ , integrating any segment of these curves will generate a linear term during a renormalization-group flow.
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