FIG. 6: (color online) The contour plots of the FLEX result at the lowest Matsubara frequency for the honeycomb lattice ( t ′ = Δ = 0) in the hexagonal Brillouin zone with n = 1 . 08 and T = 0 . 01 t (a) The Green’s function of the upper band squared, (b) the spin susceptibility, (c) the superconducting gap of the upper band, (d) the Fermi surface (the two circles) and the sign of the gap function are schematically shown in the extended zone scheme. The dashed arrows represent the wave vectors at which the spin fluctuations develop. FIG. 5: (color online) (a) The maximum value of the spin sus- ceptibility (blue, left axis) and the eigenvalue of the Eliash- berg equation (red, right) as functions of temperature for the square lattice with n = 0 . 7 (dashed) or n = 0 . 65 (dash dotted) and for the honeycomb lattice (solid) for n = 1 . 08. U = 6 t in all cases. HC and SQ stand for the honeycomb (with t ′ = 0) and the square lattices, respectively. The tempera- ture at which λ = 1 is the T c . (b) T c as functions of the level offset Δ obtained by FLEX+Eliashberg equation for the square lattice with U = 6 t , n = 0 . 7 (dashed) or for the HC lattice with U = 6 t and n = 1 . 08 (solid). which the spin susceptibility is maximized. In contrast to the case of the square lattice, one of the nodes of the d -wave gap does not intersect the Fermi surface because of its disconnectivity, i.e., the gap is like “ p -wave” if we focus only on one of the Fermi surfaces. Since smaller number of gap nodes on the Fermi surface is favorable for superconductivity, 29 this can be a reason for high T$_{c}$ despite the low DOS and the weak spin fluctuations. We now introduce the M-N level offset Δ as in the model for β -MNCl. As shown in Fig.5(b), we find that superconductivity is relatively robust against the intro- duction of Δ. This again is in sharp contrast with the square lattice, where finite Δ strongly suppresses super- conductivity. The weak effect of Δ may be because the DOS of the HC lattice is small near the band center, so that the introduction of Δ, which opens up a gap at the center of the band, is less effective than in the case of the square lattice, where the DOS diverges at the band center for Δ = 0. Let us now comment on the origin of the difference be- tween the single and two band models, namely, low T$_{c}$ triplet pairing in the former model and the high T$_{c}$ sin- glet pairing in the latter. The difference mainly comes from the difference in the band filling, i.e., for the single band model, the band filling is dilute, favoring ferromag- netic spin fluctuations, while in the two band model, it is near half filling in favor of antiferromagnetic spin fluc- there, the spin susceptibility moderately increases upon lowering the temperature. Also in Fig.5(a), we show the eigenvalue λ of the linearized Eliashberg equation as func- tions of temperature for the HC and square lattices. T$_{c}$ is the temperature where λ reaches unity. The DOS at E$_{F}$ is nearly the same for the square lattice with n = 0 . 65 and the HC lattice with n = 1 . 08, and also the spin susceptibility has similar values at low temperature, but still, the HC lattice has higher T$_{c}$ . 28 So what is the origin of this high T$_{c}$ ? Fig.6(a) shows the contour plot of the Green’s function squared, whose ridge corresponds to the Fermi surface, which consists of disconnected 2D pockets centered around K and K’ points. The spin susceptibility in Fig.6(b) is maximized at wave vectors that bridge the opposite sides of each pieces of the Fermi surface. As can be seen in Fig.6(c) (and more clearly in (d)) the gap has a d -wave form, where the gap changes sign across the wave vector at  0  0.002  0.004  0.006  0.008  0.010 0  0.2  0.4  0.6  0.8 1.0  1.2  1.4  1.6 HC n=1.08 SQ n=0.70  /  t T / t c  0  2  4  6  8 0  0.005  0.01  0.015  0.02 SQ n=0.65 T/t max (a) (b)  0.4  0.6  0.8 1.0 HC n=1.08 SQ n=0.70 4