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Chiral superconductivity is characterized by the phase of the superconducting order parameter winding by multiples of 2π around the Fermi surface (FS), breaking parity and time-reversal symmetry. Chiral d-wave superconducting state (SC state) predicted for the honeycomb lattice originates from two nodal d-wave states that are degenerate by C6v symmetry of the lattice.
In the case of graphene, the electron-driven SC state is most likely realized when doped to the vicinity of the van Hove singularity (VHS). Near the VHS, a combination of the logarithmically divergent density of states and the approximate nesting of the FS strongly enhances the effect of interactions, which can lead to the emergence of a variety of ordered states at relatively high temperatures. In previous works on graphene near the VHS filling, various electronic instabilities were analyzed using the mean-field theory, random phase approximation, quantum Monte Carlo, variational and renormalization group (RG) approaches.
In this paper we employ the truncated unity functional renormalization group (TUFRG) approach with a high momentum resolution to study the competing electronic orders on the honeycomb lattice near the VHS filling with a focus on the effect of the nearest-neighbor exchange interaction. We consider the Hubbard model including the on-site repulsion U, the nearest-neighbor repulsion V and the nearest-neighbor ferromagnetic exchange coupling J and build the tentative phase diagrams in the space of the nearest-neighbor repulsion V and the doping level δ for the fixed value of U and several typical values of J. Figure shows the phase diagrams for two values of J=0 and J=0.1t. Here the color bars indicate the values of critical scales ΩC at which the corresponding transitions may occur. In the region denoted as Metal, there is no divergence of any bosonic propagator in the RG flow down to the stopping scale Ω*=1.3x10-4eV.
In the absence of the exchange coupling J and for small nearest-neighbor repulsion V, a four-sublattice spin-density-wave (SDW4) phase is generated right around the VHS, while the chiral d-wave SC (χ-dSC) emerges slightly away from it. Upon increasing V, the spin-triplet f-wave SC (fSC) becomes dominant below the VHS. In addition, the charge-density-wave (CDW), the incommensurate charge-density-wave (iCDW) and the incommensurate spin-density-wave (iSDW) orders are also found in some regions of the phase diagrams.
The most surprising one in our results is the annihilation of the chiral d-wave SC order by a weak exchange coupling. The SC phase has completely disappeared from our phase diagram upon including the exchange coupling of J=0.1t. The investigation by a fine tuning of the parameter J shows that the phase is fully suppressed by weak exchange coupling of J≈0.05t≈0.14eV. Although we are not able to give an exact value of J for graphene, our rough estimation shows that the above value of J is inside of the range of the expectation value for the exchange coupling. Thus, our results demonstrate possible destruction of the chiral SC in single layer graphene.
As a matter of fact, the unconventional SC, which was predicted theoretically more than a decade ago for graphene doped close to the VHS, has not yet been found experimentally. Our theoretical finding that demonstrates a strong suppression of the chiral d-wave SC by weak exchange coupling might help to present a key to explain the reason for a failure of the experimental effort for finding the chiral SC in single layer graphene.
Our results have been published in "Physical Review B"(103, 235150 (2021)) under the title of "Competing electronic orders on a heavily doped honeycomb lattice with enhanced exchange coupling" (https://doi.org/10.1103/PhysRevB.103.235150).