Competing electronic instabilities of extended Hubbard models on the honeycomb lattice: A functional renormalization group calculation with high-wave-vector resolution

We investigate the quantum many-body instabilities for electrons on the honeycomb lattice at half filling with extended interactions, motivated by a description of graphene and related materials. We employ a recently developed fermionic functional renormalization group scheme, which allows for highly resolved calculations of wave-vector dependences in the low-energy effective interactions. We encounter the expected anti-ferromagnetic spin density wave for a dominant on-site repulsion between electrons, and charge order with different modulations for dominant pure $n^{\text{th}}$ nearest-neighbor repulsive interactions. Novel instabilities towards incommensurate charge density waves take place when nonlocal density interactions among several bond distances are included simultaneously. Moreover, for more realistic Coulomb potentials in graphene including enough nonlocal terms there is a suppression of charge order due to competition effects between the different charge ordering tendencies, and if the on-site term fails to dominate, the semimetallic state is rendered stable. The possibility of a topological Mott insulator being the favored tendency for dominating second-nearest-neighbor interactions is not realized in our results with high momentum resolution.

Figure 1

Diagrammatic representation of the right-hand side of Eq. (3), with particle-particle (top left), crossed particle-hole (top right), and direct particle-hole (bottom) diagrams.

Figure 2

Maximal component (out of all choices for $f_n,\mathbf{l},\left\{b_i\right\})$ of the pairing channel projection of an on-site bare interaction $U=1t$ for $f_m$'s belonging to different nearest intralattice neighbor shells in real space.

Figure 3

Example discretization for the dependence on transfer momenta $\mathbf{l}$, denser where the ordering vectors are expected. Left: Mesh of $N_k=3217$ points for momentum transfers in the particle-particle channel. Right: Mesh of $N_k=3661$ points for momentum transfers in particle-hole channels, specifically the one used for a pure $V_2$ bare interaction. For pure on-site and pure $V_1$ interactions, the mesh used for momentum transfers in particle-hole channels is the same as that used in the particle-particle channel.

Figure 4

Dominant instabilities and critical scales for different bare interaction parameters.

Figure 5

Phase diagrams of Fig. 4 with a linear plot of critical scales.

Figure 6

Critical coupling strength for a pure on-site interaction with different truncations of the form-factor basis. Calculations including up to the fourth shell of form factors are costly and thus have not been computed for $U=3.6t$. The reduction in $U_C$ for truncations including further neighbors can be understood as an effect of contributions coming from higher lattice harmonics.

Figure 7

Plots of the $\mathbf{l}$ dependence of $D_{s,s}^{\left\{b_i\right\}}\left(\mathbf{l}\right)$ at the stopping scale for different $V_1/V_2$ ratios. The ordering vectors in plots 1 and 9 are still incommensurate, though very close to $K$ and $\mathrm{\Gamma }$, respectively. Since $U$ is subcritical and there is no significant enhancement of magnetic correlations, the behavior of the more physically meaningful $K_{s,s}^{\left\{b_i\right\}}\left(\mathbf{l}\right)$ is almost indistinguishable from that of the $D$ propagator above.