Unconventional superconductivity in nearly flat bands in twisted bilayer graphene

Flat electronic bands can accommodate a plethora of interaction-driven quantum phases, since kinetic energy is quenched therein and electronic interactions therefore prevail. Twisted bilayer graphene, near the so-called “magic angles”, features slow Dirac fermions close to the charge-neutrality point that persist up to high energies. Starting from a continuum model of slow but strongly interacting Dirac fermions, we show that with increasing chemical doping away from the charge-neutrality point, a time-reversal symmetry breaking, valley pseudospin-triplet, topological $p+ip$ superconductor gradually sets in, when the system resides at the brink of an antiferromagnetic ordering (due to Hubbard repulsion), in qualitative agreement with recent experimental findings. The $p+ip$ paired state exhibits quantized spin and thermal Hall conductivities, and polar Kerr and Faraday rotations. Our conclusions should also be applicable for other correlated two-dimensional Dirac materials.

Figure 1

Various cuts of the phase diagram of interacting Dirac fermions at finite temperature ($t$) and chemical potential ($\mu$, measured from the CNP), measured in units of $v_F\mathrm{\Lambda }$. Here, $g_2^t=U\mathrm{\Lambda }/\left(12v_F\right)$ is the dimensionless coupling constant, $v_F$ is the Fermi velocity, $\mathrm{\Lambda }$ is the ultraviolet momentum cutoff for slow Dirac fermions, and $U$ is the strength of the repulsive on-site Hubbard interaction. The red shaded region is occupied by an antiferromagnet and the blue region denotes $E_g$ or $p+ip$ pairing (see also Fig. 2). The region outside these shaded ones is occupied by a correlated Dirac liquid without any long-range order. The results are obtained from a renormalization group calculation summarized in Eq. (6) and the Supplemental Material [28].

Figure 2

(a) Amplitudes of antiferromagnet ($m$) and $E_g$ pairing ($\mathrm{\Delta }$), obtained by minimizing the free energy [see Eq. (7)], as a function of small to moderate chemical doping ($\mu$), measured from the CNP, at a fixed temperature $t=0.1$. All quantities are dimensionless (see text). Corresponding variation of the coupling constants, $g_{{}_j}=(U\mathrm{\Lambda }/v_{{}_F})h_j(\mu ,t)$ in these two channels, for $j=1,2$, where $h_j(\mu ,t)$ are phenomenological functions, supplied from the outset. A finite-temperature phase diagram is qualitatively similar to the ones shown in (a) for a fixed value of $t$, since the transition temperature is proportional to the amplitude of the corresponding order parameter. The temperature scale on the right vertical axis in (a) is arbitrary. The white region at finite $t$ represents the normal state (a correlated Dirac metal). The coexisting regime in (a) arises due to an SO(5) symmetry among two order parameters [41]. To construct the phase diagram for even larger $\mu$, one should account for higher gradient terms, besides the Dirac components [37].