Simplex solids in SU($N$) Heisenberg models on the kagome and checkerboard lattices

We present a numerical study of the SU($N$) Heisenberg model with the fundamental representation at each site for the kagome lattice (for $N=3$) and the checkerboard lattice (for $N=4$), which are the line graphs of the honeycomb and square lattices and thus belong to the class of bisimplex lattices. Using infinite projected entangled-pair states and exact diagonalizations, we show that in both cases the ground state is a simplex solid state with a twofold degeneracy, in which the $N$ spins belonging to a simplex (i.e., a complete graph) form a singlet. Theses states can be seen as generalizations of valence bond solid states known to be stabilized in certain SU(2) spin models.

Figure 1

The simplex solid states obtained with iPEPS ($D=14$) for two different SU($N$) Heisenberg models. The width of a bond is proportional to the magnitude of the bond energy, while blue (red dotted) bonds correspond to negative (positive) energies. (a) One of the two degenerate trimerized ground states of the SU(3) Heisenberg model on the kagome lattice. (b) One of the two quadrumerized ground states of the SU(4) Heisenberg model on the checkerboard lattice.

Figure 2

The two different simulation setups used to simulate the kagome lattice with the iPEPS method developed for the square lattice. Black circles and triangles correspond to tensors, and black lines to the connection between tensors (the physical index of a tensor is not shown). Interactions between physical sites (solid circles) are given by thick shaded lines. (a) Auxiliary tensors (white circles) are introduced to create a square lattice iPEPS. The interactions along the horizontal and vertical direction correspond to nearest-neighbor couplings on the square lattice, whereas the interactions along the diagonal are treated as next-nearest-neighbor interactions. (b) Three physical sites $A$, $B$, $C$ on the kagome lattice, each having a local dimension $d=3$, are mapped into a block site with a local dimension $\tilde{d}=27$.

Figure 3

(a)–(c) iPEPS simulation results for the SU(3) Heisenberg model on the kagome lattice as a function of inverse bond dimension $1/D$ for the two simulation setups (cf. Fig. 2). Extrapolations (dashed lines) are a guide to the eye. (a) Energy per site compared with the results from ED (plotted as a function of $1/N_s$). The estimated value in the infinite $D$ limit is $-0.829\left(1\right)$. (b) The difference in bond energies $\Delta E$ remains finite in the infinite $D$ limit, which shows that the state breaks translational invariance as illustrated in Fig. 1a. (c) Local moment which is strongly suppressed with increasing bond dimension. (d) ED results for the energy gaps to the first singlet and the first SU(3) multiplet excitations plotted as a function of the inverse number of sites $1/N_s$.

Figure 4

ED results for the connected bond energy correlations $\langle P_{ij}{P}_{kl}\rangle -\langle P_{ij}\rangle \langle P_{kl}\rangle$. The reference bond is black, and positive (negative) correlations are shown with blue (red dashed) lines. The width of the lines is proportional to the correlation function. The periodic (Wigner-Seitz) cell is shaded in grey. Left panel: Results for the SU(3) Heisenberg model for a $N_s=21$ kagome sample. Right panel: Results for the SU(4) Heisenberg model on the checkerboard lattice for a $N_s=20$ sample.

Figure 5

iPEPS simulation results for the SU(4) Heisenberg model on the checkerboard lattice as a function of inverse bond dimension. (a) Energy per site $E_s$ compared with the results from ED (plotted as a function of $1/N_s$). (b) Different bond energies in the quadrumerized state. The strong horizontal and vertical bonds ($H$/$V$) have the same energy as the strong diagonal bonds for large $D$. (c) The difference in bond energies $\Delta E$ is finite which shows that the state breaks lattice symmetries [cf. Fig. 1b]. (d) The local moment decreases rapidly with $D$ and vanishes for large $D$.