Effect of exchange interaction on electronic instabilities in the honeycomb lattice: A functional renormalization group study

The impact of local and nonlocal density-density interactions on the electronic instabilities in the honeycomb lattice is widely investigated. Some early studies proposed the emergence of interaction-induced topologically nontrivial phases, but recently, it was denied in several works including renormalization group calculations with refined momentum resolution. We use the truncated unity functional renormalization group to study the many-body instabilities of electrons on the half-filled honeycomb lattice, focusing on the effect of the exchange interaction. We show that varying the next-nearest-neighbor repulsion and nearest-neighbor exchange integral can lead to diverse ordered phases, namely, the quantum spin Hall, the spin-Kekulé, and some spin- and charge-density-wave phases. The quantum spin Hall phase can be induced by a combination of the ferromagnetic exchange and pair hopping interactions. Another exotic phase, the spin-Kekulé phase, develops in a very small region of the parameter space considered. We encounter the three-sublattice charge-density-wave phase in a large part of the parameter space. It is replaced by the incommensurate charge density wave when increasing the exchange integral. In order to reduce the computational effort, we derive the explicit symmetry relations for the bosonic propagators of the effective interaction and propose a linear-response-based approach for identifying the form factor of order parameter. Their efficiencies are confirmed by numerical calculations in our work.

Figure 1

Honeycomb lattice, its Brillouin zone, Bravais lattice vectors in the bases, and illustration of $\pi /3$ rotation. (a) Lattice structure. The $A\left(B\right)$ sublattice is indicated by red (blue) spheres. The primitive vectors of the lattice are ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, and a unit cell is shown by the part enclosed with a gray line. (b) Brillouin zone. The gray part is the irreducible region of the Brillouin zone. (c) Bravais lattice vectors ${\mathbf{R}}_m$ in the 13 plane-wave bases $f_m\left(\mathbf{p}\right)=e^{i{\mathbf{R}}_m·\mathbf{p}}$ used by us. (d) Transfer of $A,B$ sites upon $\pi /3$ rotation followed by appropriate shift.

Figure 2

(a) A schematic representation of the order parameter ${\mathrm{\Delta }}_{o{o}^{\prime }}(\mathbf{k},\mathbf{q}-\mathbf{k})$. The left diagram in the brackets stands for the order parameter. (b) The renormalization of the vertex $S_{\mu {\mu }^{\prime }}(\mathbf{k},\mathbf{q}-\mathbf{k})$. The vertex with the gray arc is the renormalized three-point vertex, while that with the white arc is the bare vertex $S_{\mu {\mu }^{\prime }}(\mathbf{k},\mathbf{q}-\mathbf{k})$.

Figure 3

(a) Mesh of $N_q=171$ points for transfer momenta within the irreducible region of the BZ in the particle-particle channel. The points are distributed more densely near the $\mathrm{\Gamma }$ point. The bosonic propagators $P\left(\mathbf{q}\right)$ are calculated for these points. (b) Mesh of $N_q=175$ points for transfer momenta within the irreducible region of the BZ in the particle-hole channel. The points are distributed more densely near the $\mathrm{\Gamma }$ and $\mathbf{K}$ points. The bosonic propagators $C\left(\mathbf{q}\right)$ and $D\left(\mathbf{q}\right)$ are calculated for these points. (c) Mesh of $N_k=8280$ points for sampling momenta used in the integration of ${\dot{\chi }}^{\mathrm{pp}}$ and ${\dot{\chi }}^{\mathrm{ph}}$. The points are distributed more densely near the Dirac points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$.

Figure 4

Dominant instabilities for different interaction parameters. The color bar indicates the value of the critical scales ${\mathrm{\Omega }}_C$ at which the corresponding transitions may occur. In the region marked SM, there is no divergence of any bosonic propagator in the RG flow down to the stopping scale ${\mathrm{\Omega }}^{*}={10}^{-4}t$. In the region marked N/A, there is a divergence of some bosonic propagator; however, different ordering tendencies coexist and compete with each other, so that a clear identification of the leading instability is not possible, and the divergent $W^{\mathrm{SC}}\left(\mathbf{Q}\right),W^{\mathrm{SPN}}\left(\mathbf{Q}\right)$, or $W^{\mathrm{CHG}}\left(\mathbf{Q}\right)$ at ${\mathrm{\Omega }}_C$ shows a mix of various instabilities.

Figure 5

Representative spin current and bond strength patterns for the quantum spin Hall phase and spin-Kekulé phase. (a) Spin current pattern in the quantum spin Hall phase. The arrows indicate the directions of the spin currents. (b) Spin bond strength pattern in the spin-Kekulé phase. The red (blue) lines represent the positive (negative) values of spin bond order parameters, and the linewidths indicate their magnitudes. A unit cell is shown as the region enclosed by the gray line.

Figure 6

Representative charge ordering patterns for the three-sublattice charge-density-wave phase. The real charge-density distribution in the phase is expressed by linear combination of these four patterns. The radius is a measure of the local charge density on each site.