ICTP2012: Innovations in Strongly Correlated Electronic Systems

Andrey Chubukov: Collective instabilities of doped graphene

Blogged by Rafael Fernandes

 Contrasting the talk given by Oscar Vafek, who looked at many-body instabilities in double-layer graphene, Andrey will talk about instabilities in (doped) single-layer graphene.  

 Glossary: SDW = spin-density wave; SC = superconductivity; CDW = charge-density wave; FS = Fermi surface; VHS = van Hove singularity; RG = renormalization group.

 Andrey explains that when the chemical potential of graphene is changed (via electron doping, for instance), the Fermi surface changes from two Dirac points to a hexagonal FS with three saddle points, corresponding to VHS. He mentions that experiments in doped graphene find a hexagonal FS, which is somewhat similar to the one expected theoretically. However, there is no consensus that this is really the VHS of the band structure.

 Andrey proposes to study the "ideal" situation of 3/8 doping. Besides the VHS, there are three nesting vectors connecting different sides of the hexagonal FS. He mentions that such a FS is stable even when one includes next-nearest neighbor hopping. Due to the presence of VHS, the particle-particle bubble acquires an extra log, and the SC instability becomes log^2. Furthermore, due to the presence of perfect nesting, the particle-hole bubble also acquires an extra log, and density wave instabilities also becomes log^2. Other exotic instabilities may also appear, but they do not have log^2. Thus, there are several many-body instabilities, and one needs to investigate in which channels there are small attractive interactions that can take advantage of the these log instabilities. Ultimately, Andrey wants to find which of the several instabilities is the leading one.

 To address this problem, he will use parquet RG, which treats particle-particle and particle-hole channels on equal footing. In the RPA level, the pairing interaction is repulsive, but SDW and CDW channels are attractive. However, these are the bare interactions, which become renormalized at low energies. Andrey now writes all the possible interactions between the low-energy fermions near the VHS in each of the three FS patches. There are four of them, including density-density interactions, pair-hopping, exchange, etc. Doing the one-loop parquet RG, these interactions are renormalized and change as higher energy modes are integrated out. The question Andrey wants to answer is which of these four coupling constants diverges, and which order it triggers. Formally, one has a set of coupled first-order differential equations. Andrey considers both the cases of perfect nesting and non-perfect nesting - the latter via an effective parameter.

 Q: is umklapp considered? A: sure, pair-hopping is an umklapp process.

 Andrey shows the solution of the flow equations - all couplings diverge at a certain energy scale, but one of them diverges first. By writing down the SC, SDW, and CDW vertices, and plugging in the renormalized couplings, he can identify which one diverges first. The two leading instabilities are the SDW (whose bare coupling was positive - attractive) and singlet SC (whose bare coupling was negative - repulsive). Eventually (i.e. at lowest energies), SC wins and becomes the leading instability. Several technical questions pop up, from the validity of the RG procedure for log^2 instability to the fate of the flow equations away from perfect nesting.

 Now the symmetry of the SC state is discussed. There are three gaps corresponding to the three VH points, and three possible eigenvectors/eigenvalues for the linearized gap equations. Two of these eigenvalues are actually degenerate - the other one, whose eigenvector corresponds to the s-wave solution, remains always negative, i.e. it is not realized. The two degenerate solutions have d-wave symmetry: d_xy and d_{x^2-y^2}. By performing a Ginzburg-Landau expansion, Andrey finds that the solution with lowest energy is actually a combination of these two symmetries: it is a (d+id) state, which breaks time-reveral symmetry. Interestingly, functional RG gives the same result.

 Conclusion: graphene doped to the saddle-point of the band structure displays as the leading instability a time-reversal (d+id) SC state!