Functional renormalization group and variational Monte Carlo studies of the electronic instabilities in graphene near $\frac{1}{4}$ doping

We study the electronic instabilities of near 1/4 electron doped graphene using the singular-mode functional renormalization group, with a self-adaptive $k$ mesh to improve the treatment of the van Hove singularities, and variational Monte Carlo method. At 1/4 doping the system is a chiral spin-density wave state exhibiting the anomalous quantized Hall effect. When the doping deviates from 1/4, the $d_{x^2-y^2}+i{d}_{xy}$ Cooper pairing becomes the leading instability. Our results suggest that near 1/4 electron or hole doping (away from the neutral point) the graphene is either a Chern insulator or a topoligical superconductor.

Figure 1

A generic 4-point vertex (a) is rearranged into the pairing (b), crossing (c) and direct (d) channels. Here $\mathbf{k},\mathbf{q},\mathbf{p}$ are momenta, $\sigma$ and $\tau$ denote spins which are conserved during fermion propagation, and $m,n$ denote the form factor (see the text for details). The open arrows indicate collective propagation.

Figure 2

Results for $\delta =1/4$ and $V=0$. (a) The Fermi surface. The brightness on the surface represents the momentum space gap function associated with one of the degenerate pairing modes. (b) FRG flow of (the inverse of) the most negative singular values $S$ in the SC (blue solid line), SDW (green dot-dashed line), and CDW (red dashed line) channels. (c) and (d) are the renormalized interaction $V_{sdw}^{mm}$ for $m=(s_0,A)$, and $V_{sc}^{mm}$ for $m=(d_1,A)$, as functions of the collective momentum $\mathbf{q}$. The hexagons in (a), (c), and (d) indicate the Brillouin-zone boundary.

Figure 3

Chiral SDW order on the honeycomb lattice. (a) The spins on the black, red, green, and blue sublattices (of different gray scales) are ${\mathbf{M}}_1+{\mathbf{M}}_2+{\mathbf{M}}_3$, $-{\mathbf{M}}_1-{\mathbf{M}}_2+{\mathbf{M}}_3$, ${\mathbf{M}}_1-{\mathbf{M}}_2-{\mathbf{M}}_3$, $-{\mathbf{M}}_1+{\mathbf{M}}_2-{\mathbf{M}}_3$ respectively. (b) A three-dimensional perspective view of the chiral SDW order.

Figure 4

Variational Monte Carlo results for the energy gain per site, $\Delta E$, versus the variational order parameters $\Delta$ for the $d+i{d}^{\prime }$-SC (circles and triangles) and the chiral SDW states (squares) at the doping level $\delta =1/4$. The lattice sizes are given in the legends. Lines are guides to the eye.

Figure 5

The same plots as in Fig. 2 but for $\delta =0.211$. Note the splitting of the SDW peaks in panel (c) signifies the incommensurate SDW instability.

Figure 6

(a) The FRG diverging energy scale ${\Lambda }_c$ plotted as a function of doping level near $\delta =1/4$. Crosses and open circles represent ${\Lambda }_c$ associated with the SC and SDW channel, respectively. $V=0,t_1$ for solid and dashed lines, respectively. (b) A schematic temperature-doping phase diagram near $\delta =1/4$ in linear scales. The gray region denotes the transition between SC and SDW.

Figure 7

One-loop diagrams contributing to the flow of the four-point vertex function in the pairing channel (a), crossing channel (b), and direct channel (c)–(e). Here $m,m^{\prime }n,n^{\prime }$ denote form factors, while the momentum and spin indices are left implicit. The open arrows indicate the flow of the collective momentum. The slashed lines are single-scale fermion propagators. The slash can be placed on either internal line associated with the loop.

Figure 8

Examples of self-adaptive $\mathbf{k}$-mesh (a) and $\mathbf{q}$-mesh (b) used for loop integrations and collective momenta for interactions, respectively. Notice that in (a) the mesh is too dense near the Fermi surface to be differentiated by the naked eye.