Superconductivity on the Brink of Spin-Charge Order in a Doped Honeycomb Bilayer

Using a controlled weak-coupling renormalization group approach, we establish the mechanism of unconventional superconductivity in the vicinity of spin or charge ordered excitonic states for the case of electrons on the Bernal stacked bilayer honeycomb lattice. With one electron per site, this system, physically realized in bilayer graphene, is unstable towards a spontaneous symmetry breaking. Repulsive interactions favor excitonic order, such as a charge nematic and/or a layer antiferromagnet. We find that upon adding charge carriers to the system, the excitonic order is suppressed, and unconventional superconductivity appears in its place, before it is replaced by a Fermi liquid. We focus on firmly establishing this phenomenon using the renormalization group formalism within an idealized model with parabolic touching of conduction and valence bands.

Figure 1

(a) The flow of interaction couplings $g_j$ defined in Eq. (1), when initially $g=g_{A_{1g}}(0)>0$, while all other couplings vanish. Under such conditions, only $g_{A_{2g}}$ and $g_{E_g}$ get generated, while all other couplings vanish. Here $\mu =0$. (b) The corresponding flow of the Cooper pair couplings ${\tilde{g}}_j$ defined in Eq. (5). All nine ${\tilde{g}}_j$’s are finite as can be seen from the Fierz matrix (8); their values are independent of the $g$,$u$, and $\mathbf{K}$ label, i.e., ${\tilde{g}}_{A_{1g}}={\tilde{g}}_{A_{1u}}={\tilde{g}}_{A_{1\mathbf{K}}}={\tilde{g}}_{A_1}$, etc. Note that if $g$ is small then the attractive interactions are generated in the regime where the weak coupling RG is fully justified.

Figure 2

RG flows with $g_{A_{1g}}(0)=0.05\mathrm{vs}$ $t$ [Eq. (10)]. For the solid curves $\mu ={10}^{-9}{\mathrm{\Omega }}_{\mathrm{\Lambda }}$; for the dashed curves $\mu =3\times {10}^{-6}{\mathrm{\Omega }}_{\mathrm{\Lambda }}$. The former is in the regime $\mu \ll {\mathrm{\Omega }}_{\mathrm{\Lambda }}{e}^{-2C_{*}/g}$ leading to an excitonic order, the latter in ${\mathrm{\Omega }}_{\mathrm{\Lambda }}{e}^{-2C_{*}/g}\ll \mu \ll {\mathrm{\Omega }}_{\mathrm{\Lambda }}{e}^{-2C_1/g}$ leading to SC. (a) $\tilde{g}$ in Eq. (5). (b) Susceptibilities in excitonic and superconducting channels.

Figure 3

Coupling flows for various values of bare couplings, with $\mu =0$. $\lambda =0$ corresponds to the forward-scattering limit, in which only $g_{A_{1g}}$ is nonzero, while $\lambda =1$ corresponds to on-site Hubbard interaction.

Figure 4

Temperature $T^{*}$ associated with ordering versus $\mu$ and $g$ for the forward-scattering limit. The crossover lines shown correspond to asymptotic bounds for the phase boundaries between excitonic (E), superconducting (SC), and Fermi liquid (FL) states.