Dimerized Solids and Resonating Plaquette Order in $\mathrm{SU}(N)$-Dirac Fermions

We study the quantum phases of fermions with an explicit $\mathrm{SU}(N)$-symmetric, Heisenberg-like nearest-neighbor flavor exchange interaction on the honeycomb lattice at half filling. Employing projective (zero temperature) quantum Monte Carlo simulations for even values of $N$, we explore the evolution from a weak-coupling semimetal into the strong-coupling, insulating regime. Furthermore, we compare our numerical results to a saddle-point approximation in the large-$N$ limit. From the large-$N$ regime down to the SU(6) case, the insulating state is found to be a columnar valence bond crystal, with a direct transition to the semimetal at weak, finite coupling, in agreement with the mean-field result in the large-$N$ limit. At SU(4) however, the insulator exhibits a subtly different valence bond crystal structure, stabilized by resonating valence bond plaquettes. In the SU(2) limit, our results support a direct transition between the semimetal and an antiferromagnetic insulator.

Figure 1

Ground state phase diagram of fermions with $\mathrm{SU}(N)$-symmetric flavor exchange on the honeycomb lattice. Crosses denote parameters at which QMC simulations have been carried out. For all considered (even) $N$ the system undergoes a quantum phase transition from a semimetal (SM) to an insulator. For $N\ge 6$ the insulating state is a columnar valence bond solid (cVBS), while at $N=4$ it is a valence bond solid with resonating valence bond plaquettes (pVBS); both are depicted in the right inset. At $N=2$ an antiferromagnetic insulator (AFM) appears. The left inset shows the lattice structure with the six-sites unit cell employed in the large-$N$ calculations.

Figure 2

$N=10$. Finite size extrapolation of the AFM structure factor $S_{\mathrm{AFM}}(\mathbf{\Gamma })$, the spin gap ${\Delta }_{\sigma }(\mathbf{\Gamma })$, and the single-particle gap ${\Delta }_{\mathrm{sp}}(\mathbf{K})$ in the VBS for SU(10) at $J/t=6.0$. The inset shows the finite size extrapolation of the structure factors $S_{\dim }(\mathbf{K})$ and $S_{\mathrm{kin}}(\mathbf{K})$.

Figure 3

$N=10$. Real-space dimer correlations $D_{ij,kl}$ on a $L=6$ lattice for SU(10) at $J/t=6.0$, with the striped reference bond. $D_{ij,kl}$ is indicated by colors and the line thickness in addition to the explicit values given for each bond.

Figure 4

$N=4$. Finite size scaling of the dimer and kinetic structure factors $S_{\dim }(\mathbf{K})$ and $S_{\mathrm{kin}}(\mathbf{K})$ at SU(4).

Figure 5

Histograms of the dimer density along three bond types (illustrated in the top left panel), which correspond to three realizations of the same columnar VBS. Predominant weight at the corners in the phase space represent the columnar states as seen at $N=10$ and 6. A resonating plaquette is formed on two different bond types and is signaled by weight at the midpoint between two bond directions as realized at SU(4). All histograms have been rescaled in order to ease the identification of their characteristic distributions.

Figure 6

$N=2$. Staggered magnetization $m_s=\sqrt{S_{\mathrm{AFM}}(\mathbf{\Gamma })}/L$ and single-particle gap ${\Delta }_{\mathrm{sp}}(\mathbf{K})$ as functions of the system size for SU(2) and different ratios $J/t$. The inset shows the finite size scaling of the structure factor $S_{\dim }(\mathbf{K})$ inside the range $2.1\lesssim J/t\lesssim 2.5$.