Competing nematic, antiferromagnetic, and spin-flux orders in the ground state of bilayer graphene

We analyze the phase diagram of bilayer graphene (BLG) at zero temperature and zero doping. Assuming that at high energies the electronic system of BLG can be described within a weak-coupling theory (consistent with the experimental evidence), we systematically study the evolution of the couplings with going from high to low energies. The divergences of the couplings at some energies indicate the tendency towards certain symmetry breakings. Carrying out this program, we found that the phase diagram is determined by microscopic couplings defined on the short distances (initial conditions). We explored all plausible space of these initial conditions and found that the three states have the largest phase volume of the initial couplings: nematic, antiferromagnetic, and spin flux (a.k.a. quantum spin Hall). In addition, ferroelectric and two superconducting phases appear only near the very limits of the applicability of the weak-coupling approach. The paper also contains the derivation and analysis of the renormalization group equations and the group theory classification of all the possible phases which might arise from the symmetry breakings of the lattice, spin rotation, and gauge symmetries of graphene.

Figure 1

Left panel: 3D view of bilayer graphene. The sites that sit on top of each other, connected by dotted lines, hybridize strongly and form bands with a gap of ${\gamma }_1\approx 0.4$ eV. The low-energy electrons live on the half of the carbon atoms that sit over/under the centers of the hexagons. Right panel: Top-down view of the lattice.

Figure 2

Sketches of the density and currents transforming according representations of the group ${\mathcal{D}}_{3d}^{\prime \prime }$. In the case of a spin singlet symmetry breaking, the plus and minus signs represent charges, the blue lines represent persistent currents, and the black bars represent bonds. The $G$, $E_2$, and $E_2^{\prime \prime }$ order parameters triple the unit cell; the new Bravais lattice vectors are given by the dashed arrows. The representations are given in terms of Pauli matrices in Eqs. (2.11) and (2.10).

Figure 3

Four cuts through the possible parameter space of BLG. The predicted gap (or saddle point energy for only gapless nematic phase) is indicated by the color scale and the predicted phase is indicated. N is a nematic, AF is an antiferromagnetic phase, SF is a spin flux phase, and FE is a ferroelectric. A fifth predicted superconducting phase is in a range parameters not shown; see Fig. 9 for more details. The $g$ are coupling constants of BLG, with the subscript labeling the irreducible representations in accordance with Fig. 2 and defined in Sec. 2. All boundaries are the first-order phase transitions.

Figure 4

Definition of the elements of the diagrammatic expansion. The thick line is the fermion propagator; the circle is the self-energy from the single particle of the spectrum. The wavy line is the Coulomb propagator and the dotted line is the contact interaction. We separate the scalar contact interaction $g_{00}$ from the other interactions.

Figure 5

Resummation of the strong Coulomb interaction in the $1/N$ approximation. (a) Evaluation of the polarization loop. (b) Definition of the resummed propagator, represented by the double wavy line. The scalar contact interaction is included in the resummation as it has the same matrix structure as the Coulomb interaction. (c) The non-renormalization of the Coulomb vertex as a result of gauge invariance. [$\delta Z$ is defined in Fig. 6a.]

Figure 6

Diagrams included in RG equations. (a) Single-particle particle weight renormalization. (b) Renormalization of the mass $m$. (c) Renormalization of the contact interactions.

Figure 7

(a) Schematic representation of the “bubble” diagrams where the shaded blob represents all possible connected diagrams, and arbitrary Coulomb propagators may be added. (b) Similar representation of the “ladder” diagrams. The leading diagrams from both of these groups are included, even though this is not strictly parametrically correct. (c) A second loop contribution to the anomalous dimension of the coupling $g_{E_2}$ which is disregarded.

Figure 8

Plots of the coupling constant as a function the running RG scale $\ell =\ln (K_0/K)$. There are eight running couplings labeled by the corresponding representation. The density-density couplings are given by solid lines, the current-current couplings by dashed. The graphs end when the couplings become of order $1/N=1/4$. They reach a singularity a finite $\ell$ soon after the graph ends.

Figure 9

Plot of the broken symmetry phase and energy scale for BLG as a function of bare coupling constants. When the energy scale is less than $E_{\mathrm{LiTr}}\approx 1$ meV the broken symmetry state is in competition with the Lifshitz transition and the BLG may remain metallic (with eight Dirac points). Dashed line indicates the region where the triplet superconducting phase (SC) is very close in energy (but slightly above) to normal (nematic or ferroelectric) states.

Figure 10

Interaction energy as a function of energy scale for selected phase with initial conditions $g_G=-0.0$, $g_{B_2}=0.06$, $g_{E_2^{\prime \prime }}=0$, $g_{E_2}=-0.03$, and all others zero. For these initial conditions the nematic phase is the ground state.

Figure 11

The interaction energy as a function of energy scale for some choice of initial parameters with initial conditions $g_G=-0.075$, $g_{B_2}=0.06$, $g_{E_2^{\prime \prime }}=0$, $g_{E_2}=0.015$, and all others zero. In this case the ground state is the AF phase.

Figure 12

Interaction energy for selected phases a function of energy scale with initial conditions $g_G=0$, $g_{B_2}=0.06$, $g_{E_2^{\prime \prime }}=0.06$, $g_{E_2}=0.03$, and all others zero. In this case the ground state is the spin flux state shown in orange.

Figure 13

The evolution of the interaction energy of the selected phases as a function of the energy scale with initial conditions $g_G=0.06$, $g_{B_2}=0$, $g_{E_2^{\prime \prime }}=0.06$, $g_{E_2}=0.075$, and all others zero. In this case the ground state is a ferroelectric phase. The difference in energy between the F phase and nearby phases is quite small.

Figure 14

Interaction energy for selected phases as a function of the energy scale with initial conditions $g_G=-0.045$, $g_{B_2}=0.06$, $g_{E_2^{\prime \prime }}=-0.06$, $g_{E_2}=0.09$, and all others zero. In this case the singlet superconducting phase is the ground phase.

Figure 15

Interaction energy for selected phases as a function of the energy scale for initial repulsive short-range interaction. In this case the triplet superconducting phase is very close to the ground phase with initial conditions $g_G=0.06$, $g_{B_2}=0.06$, $g_{E_2^{\prime \prime }}=-0.12$, $g_{E_2}=0.06$, and all others zero. Even though $c_{\mathrm{SC}}$ appears to be the most negative, minimization of Eq. (4.13) gives the nematic state as a preferred ground state.

Figure 16

Sketch of the symmetric magnetic and normal phases transforming according the $G$ representation. (I) $G_1$ normal state; (II) $G_2$ normal state; (III) $G_3$ spin state; (IV) $G_4$ spin state; (V) $G_5$ spin state. Because of the absence of spin-orbit coupling the overall direction of the spins is arbitrary.