Superconductivity from weak repulsion in hexagonal lattice systems

We analyze the pairing instabilities for fermions on hexagonal lattices (both honeycomb and triangular ones) in a wide range of fermionic densities ranging from van Hove density at which a single large Fermi surface splits into two disconnected Fermi pockets, to a density at which disconnected pockets shrink to Fermi points (half-filling for a honeycomb lattice and full filling for a triangular lattice). We argue that for a generic doping in this range, superconductivity at weak coupling is of Kohn-Luttinger type, and, due to the presence of electronic interactions beyond on-site repulsion, is a threshold phenomenon, with superconductivity emerging only if the attraction generated by the Kohn-Luttinger mechanism exceeds the bare repulsion in some channel. For disconnected Fermi pockets, we predict that Kohn-Luttinger superconductivity, if it occurs, is likely to be $f$ wave. While the Kohn-Luttinger analysis is adequate over most of the doping range, a more sophisticated analysis is needed near van Hove doping. We treat van Hove doping using a parquet renormalization group, the equations for which we derive and analyze. Near this doping level, superconductivity is a universal phenomenon, arising from any choice of bare repulsive interactions. The strongest pairing instability is into a chiral $d$-wave state ($d+id$). At a truly weak coupling, the strongest competitor is a spin-density-wave instability, however, $d$-wave superconductivity still wins. Moreover, the feedback of the spin density fluctuations into the Cooper channel significantly enhances the critical temperature over the estimates of the Kohn-Luttinger theory. We analyze renormalization group equations at stronger couplings and find that the main competitor to $d$-wave superconductivity away from weak coupling is actually ferromagnetism. We also discuss the effect of the edge fermions and show that they are unimportant in the asymptotic weak coupling limit, but may give rise to, e.g., a charge-density-wave order at moderate coupling strengths.

Figure 1

Kohn-Luttinger mechanism of the pairing. The irreducible pairing interaction is the sum of the Coulomb interaction, which includes all regular contribution from screening [$U\left(q\right)$, represented by a dashed line], and nonanalytic terms, which appear at second order in $U\left(q\right)$. Kohn and Luttinger demonstrated that for rotationally isotropic systems, the partial components of the irreducible pairing interaction are attractive for arbitrary $U\left(q\right)$ for large values of the angular momentum $l$, at least for odd $l$.

Figure 2

Fermi surface (blue lines) for fermions on a honeycomb lattice at van Hove doping ($\nu =3/8$ or $5/8$, where $\nu =1/2$ corresponds to half-filling, where the fermionic spectrum has Dirac points). For nearest-neighbor hopping, the Fermi surface consists of parallel pieces (nesting). At the end point of parallel pieces the density of states diverges (van Hove points $M_1,M_2$, and $M_3$). Thin solid lines represent the boundaries of the Brillouin zone.

Figure 3

The flow of the couplings in the $s$-wave superconducting (SC) channel and spin-density-wave (SDW) channel under parquet RG at van Hove density. The SDW coupling wins at intermediate scales. The SC coupling is initially repulsive, but changes sign under RG flow and eventually wins over the coupling in the SDW channel.

Figure 4

Triangular lattice (a) and the corresponding Brillouin zone in momentum space (b).

Figure 5

Evolution of the FS with doping for fermions on a triangular lattice: (a) $\mu$ slightly above $-8t$, (b) $\mu$ approaching zero, (c) $\mu =0$. This is van Hove doping. The FS consists of parallel pieces, which end at van Hove points where the density of states diverges. (d) $\mu <t$. The FS consists of disconnected pieces.

Figure 6

The vertex function, whose poles determine the pairing instability, is the fully renormalized and antisymmetrized interaction at zero total incoming and outgoing momenta.

Figure 7

The pairing vertices for spin-singlet and spin-triplet channels.

Figure 8

Patch model I assumes that the superconducting gap is concentrated in the shaded patches (a) van Hove doping. The patches are centered at van Hove points: (b) and (c) above and below van Hove doping, respectively; (d) for doping line in (c), the model allows for symmetric and antisymmetric solutions between points 1 and $\overline{1}$.

Figure 9

Patch model II. At van Hove doping (a) the patches are centered in between van Hove points: (b) and (c) above and below van Hove doping, respectively. For a FS in the form of disconnected pieces (c), the patches are located at the centers of FS arcs.

Figure 10

(a) and (b) Structure of the superconducting gap in $s$-, $p$-, $d$-, and $f$-wave channels at van Hove doping in patch models I and II, respectively. In the $p$-wave and $d$-wave channels, the eigenvalues are doubly degenerate. (c) The structure of the gaps in spin-triplet channels in patch model I at $\mu <0$, when the FS is singly-connected and does not reach van Hove points.

Figure 11

The structure of superconducting gaps in $s$-wave, $f$-wave, and two degenerate $d$-wave and $p$-wave channels for the symmetric solution for the patch model I for $\mu >0$, when the FS consists of six disconnected pieces. About the same solutions are obtained in patch model II in $s$-wave and $p$-wave channels.

Figure 12

The same as in Fig. 11, but for the antisymmetric solution for the patch model I. About the same solutions are obtained in patch model II in $f$-wave and $d$-wave channels.

Figure 13

The structure of the gap in $s$-, $p$-, $d$-, and $f$-wave channels in patch model II at $0<\mu <t$, when the FS consists of six disconnected segments.

Figure 14

(a) and (b) The coupling constants in $f$-wave and in $d$-wave channels as functions of the chemical potential for fermions on a triangular lattice. (c) The transition temperatures for $f$-wave and chiral $d+id$ superconductivity. The coupling in $f$-wave channel remains finite and positive at $\mu =t$, when the FS disappears, but $T_c$ for $f$-wave vanishes because of vanishing prefactor. Near van Hove doping, $f$-wave channel is already repulsive. As a result, $T_c$ for $f$ wave is peaked in between $\mu =0$ and $\mu =t$. The degenerate $d$-wave channel has the largest coupling at van Hove doping, but becomes repulsive at $\mu \approx t$. Whether the attractive regions in $d$-wave and $f$-wave channels overlap depends on the interplay between KL terms and the first-order terms coming from interaction between first, second, and, possibly, further neighbors. Moreover, at intermediate $\mu$, SDW well may become the leading instability.

Figure 15

A honeycomb lattice (a) and the corresponding Brillouin zone in momentum space (b).

Figure 16

Scaling function $f\left(q/k_F\right)$ in Eq. (52). The limiting values are $f\left(0\right)=1$ and $f\left(2\right)=3.66/\pi$.

Figure 17

An example of higher-order diagram, which contributes to KL renormalization of the pairing interaction near half-filling. Thin blue and thick red lines represent fermions from the two branches with positive and negative energies, respectively. The dashed line is the Hubbard interaction $U\left(0\right)$.

Figure 18

Fermions on a honeycomb lattice. (a) and (b) Behavior of the leading eigenfunctions in $f$-wave and $d$-wave channels as functions of the chemical potential for $0<\mu <t$, when the FS consists of six disconnected segments. (c) Behavior of $T_c$ in the two channels. The $d$-wave channel has the strongest instability (largest $T_c$) near van Hove doping.

Figure 19

(a) Shows hexagonal FS inscribed within hexagonal BZ. The three inequivalent corners of the FS, $M_1,M_2,M_3$ are saddle points of the dispersion and dominate the density of states. The FS displays three nesting vectors $Q_1,Q_2$ and $Q_3$. (b) The FS may be split into corner regions $A$, $B$, and $C$ and edge regions $\pm 1,\pm 2,\pm 3$.