Plaquette order in the SU(6) Heisenberg model on the honeycomb lattice

We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et al., Phys. Rev. A 84, 011611(R) (2011)]. Using exact diagonalizations of finite clusters, infinite projected entangled pair state simulations, and variational Monte Carlo simulations based on Gutzwiller projected wave functions, we provide strong evidence that the model with one particle per site and nearest-neighbor exchange actually develops plaquette order. This is further confirmed by the investigation of the model with a ring-exchange term, which shows that there is a transition between the plaquette state and the chiral state at a finite value of the ring-exchange term.

Figure 1

(a) Energies of Gutzwiller projected wave functions and (b) the bond energies on $t_d$ and $t_h$ bonds after projection for the different flux configurations as a function of $t_d/t_h$ for $N_s=72$. (c)–(e) shows the considered flux configurations, where black bonds represent hopping amplitude $t_d$, while dark and light purple bonds denote hopping amplitudes $t_h$ and $-t_h$, respectively. In the case of the uniform $2\pi /3$ flux configuration, red arrows represent complex hopping amplitude $\propto e^{i2\pi /3}$, for which $t_{ji}=t_{ij}^{*}$.

Figure 2

Spectrum of the 18-site (left) and 24-site (right) clusters as a function of the quadratic Casimir ${\mathcal{C}}_2$. The degeneracies of some states are indicated, as well as the spatial quantum numbers for the 18-site cluster. For the 24-site cluster, the presence of three low-lying states is a strong indication of a plaquette phase (see text for details). Inset: Broken-symmetry plaquette state reconstructed from ED. It breaks translations, but the $D_6$ symmetry is preserved. The bond energy is $-0.81 (-0.56$) for the thick (thin) lines.

Figure 3

(a) $〈P_{0i}〉-1/6$ spin-spin and (b) $〈P_{12}{P}_{kl}〉-〈P_{12}〉〈P_{kl}〉$ dimer-dimer correlations in the exact ground state of the 24-site cluster. As a reference, we present the dimer-dimer correlations of the translational invariant linear combination of (c) the variational $0\pi \pi$ flux projected states with $\left|t_d/t_h\right|=0.8$, and of (d) the variational $2\pi /3$ flux projected state. The pattern of the dimer-dimer correlations of (b) the ED and (c) the $0\pi \pi$ variational states is an indication of long-range plaquette ordering.

Figure 4

iPEPS results for the SU(6) Heisenberg model on the honeycomb lattice. (a) Comparison of the ground state energy obtained with iPEPS, VMC, and ED, as a function of inverse bond dimension $D$ and inverse system size. The bold symbols mark improved VMC results for $N_s=24$ and 72 (see Table 1 and main text). For large bond dimension with iPEPS the plaquette state has the lowest variational energy, in agreement with VMC. (b) Color-order parameter as a function of inverse $D$. It is finite for the color-ordered state and vanishes for the plaquette state. (c) Difference in energy between the strongest bond and the weakest bond in the unit cell, which is strongly suppressed in the color-order state, and finite in the plaquette state, consistent with plaquette long-range order. The dotted lines are a guide to the eye.

Figure 5

Comparison of the ED spectrum (black points) of the model of Eq. (3) with the variational energies (continuous lines) based on Gutzwiller projected wave functions for the $0\pi \pi$ plaquette phase and the $2\pi /3$ chiral phase. The inset shows the (ordered) eigenvalues ${\lambda }_j$ of the overlap matrices of the projected states with different twisted boundary conditions before projection as a function of $j$.