Electronic instabilities of the extended Hubbard model on the honeycomb lattice from functional renormalization

Interacting fermions on the half-filled honeycomb lattice with short-range repulsions have been suggested to host a variety of interesting many-body ground states, e.g., a topological Mott insulator. A number of recent studies of the spinless case in terms of exact diagonalization, the infinite density matrix renormalization group, and the functional renormalization group, however, indicate a suppression of the topological Mott insulating phase in the whole range of interaction parameters. Here, we complement the previous studies by investigating the quantum many-body instabilities of the physically relevant case of spin-1/2 fermions with onsite, nearest-neighbor, and second-nearest-neighbor repulsion. To this end, we employ the multipatch functional renormalization group for correlated fermions with refined momentum resolution observing the emergence of an antiferromagnetic spin-density wave and a charge-density wave for dominating onsite and nearest-neighbor repulsions, respectively. For dominating second-nearest neighbor interaction our results favor an ordering tendency towards a charge-modulated ground state over the topological Mott insulating state. The latter evades a stabilization as the leading instability by the additional onsite interaction.

Figure 1

(Left panel) Lattice structure in real space. The two different sublattices $A$ and $B$ are indicated by empty and filled circles, respectively, connected by the nearest-neighbor vectors ${\vec{\delta }}_1,{\vec{\delta }}_2,{\vec{\delta }}_3$. The second-nearest-neighbor vectors are given by ${\vec{\mathrm{\Delta }}}_1,{\vec{\mathrm{\Delta }}}_2,{\vec{\mathrm{\Delta }}}_3$ and the bond dimerization corresponding to the QAH and QSH ordering patterns is indicated by ${\chi }_{ij}$; see, for example, Ref. [4] for a discussion of the mean-field order parameters. (Right panel) Nearest-neighbor hopping on the honeycomb lattice provides the depicted energy dispersion with the conduction (blue) and the valence band (red) touching linearly at the $K,K^{\prime }$ points at the Fermi level.

Figure 2

(Left panel) Sketch of the BZ with patches and patch points with $N_a=4$ and $N_r=4$. (Right panel) Numbering of angular and radial patches according to the patching scheme.

Figure 3

Critical interaction strength $U_c$ as a function of the number of patch rings. $N_a$ and $N_r$ denote the number of angular and radial patches, respectively. We observe a smooth convergence in the value for $U_c$ for $N_a=6$. In the following instability analysis we therefore use a patching resolution with $N_{\text{tot}}=6N_a{N}_r=144$.

Figure 4

Vertex signature for the CM or ${\mathrm{CDW}}_3$ instability. (Left and middle panel) Effective coupling in units of $t$ as a function of the incoming patch indices $\pi \left({\vec{k}}_1\right)$ (vertical) and $\pi \left({\vec{k}}_2\right)$ (horizontal) with the third patch index $\pi \left({\vec{k}}_3\right)$ fixed at the patch point with $N_a=1$ in the left lower corner of the BZ; cf. Fig. 2. For simplicity, the vertex structure is only shown for patch points on the second ring. The two vertex pictures correspond to different sublattice combinations of in- and outgoing momenta. (On the left) $o_1=o_2=o_3=o_4$. (In the middle) $o_1=o_3\ne o_2=o_4$. The calculations were performed with $N_a=6$ and $N_r=4$. The large components in this interaction vertex are used to extract the effective form of the Hamiltonian given in Eq. (8). (Right panel) Qualitative charge distribution of the CM mean-field state for $\alpha =\pi /3$.

Figure 5

Tentative phase diagrams for spin-1/2 fermions on the honeycomb lattice at half-filling as a function of the short-range repulsions $U,V_1$, and $V_2$ in units of the hopping amplitude $t$. The different panels show different values of the onsite interaction $U/t=0, U/t=1, U/t=2$, and $U/t=3$. The numerical evaluation of the multipatch RG flow equations was performed with a high-definition patching resolution of $N_a=6$ and $N_r=4$ corresponding to a total number of 144 patches in the BZ. In the region marked with N/A, where different ordering patterns or the semimetallic state meet, a clear identification of the leading instability was not possible as the vertex close to ${\mathrm{\Lambda }}_c$ did not show one clear signature but rather a mix of various instabilities.

Figure 6

Tentative phase diagram for the different sets of short-ranged interaction parameters $U,V_1,V_2$ following a Coulomb-inspired shape of the interaction potential, Eq. (11), with onsite repulsion strength $U_0$ and screening parameter $\epsilon$. Again, we evaluate the flow equations with a patching resolution of 144 patches in the BZ.