Simplex valence-bond crystal in the spin-1 kagome Heisenberg antiferromagnet

We investigate the ground-state properties of a spin-1 kagome antiferromagnetic Heisenberg model using tensor-network (TN) methods. We obtain the energy per site $e_0=-1.41090\left(2\right)$, with $D^{*}=8$ multiplets retained (i.e., a bond dimension of $D=24)$, and $e_0=-1.4116\left(4\right)$ from large-$D$ extrapolation, by accurate TN calculations directly in the thermodynamic limit. The symmetry between the two kinds of triangles is spontaneously broken, with a relative energy difference of $\delta \approx 19\%$, i.e, there is a trimerization (simplex) valence-bond order in the ground state. The spin-spin, dimer-dimer, and chirality-chirality correlation functions are found to decay exponentially with a rather short correlation length, showing that the ground state is gapped. We thus identify the ground state to be a simplex valence-bond crystal. We also discuss the spin-1 bilinear-biquadratic Heisenberg model on a kagome lattice, and determine its ground-state phase diagram. Moreover, we implement non-Abelian symmetries, here spin SU(2), in the TN algorithm, which improves the efficiency greatly and provides insight into the tensor structures.

Figure 1

(a) Kagome lattice (dotted lines) and the initial setup of the tensor-network wave function (solid lines). $D$ is the bond dimension, and $T_A$ $\left(T_B\right)$ are triangle tensors, with which the physical indices can be associated for convenience. (b) Illustration of the simplex valence-bond crystal. The two kinds of triangles or “simplexes” [of type $A$ (blue) and $B$ (pink)] have different energies, and a lattice inversion symmetry is spontaneously broken.

Figure 2

The variational ground-state energy per site $e_0$ is shown vs $1/D$, obtained from iPEPS contractions [with and without implementing SU(2) symmetry] on the infinite kagome lattice, using both ST and DT update schemes. The inset shows that the $D\ge 12$ data (i.e., left-hand side of the dashed line) converge exponentially to the infinite $D$ limit, which is extrapolated as $e_0^{\infty }=-1.4116\left(4\right)$. The convergence of $e_0$ vs truncation parameters $d_c$ have always been checked (see the Supplemental Material [36]), and the data above are obtained with $d_c=40–60$ and $100–120$ for plain and SU(2) iPEPS contractions, respectively.

Figure 3

(a) Illustration of the cylinders. For XC (YC) geometries, X (Y) direction is with the periodic boundary condition, and length unit $a_x\left(a_y\right),V$ is the boundary vector obtained by exact contractions of cylinder PEPS. (b) Implementation of SU(2) symmetry in local tensors: The arrows indicate how the spin multiplets are fused together [40]. The table in (c) shows the specific spin representations $Q_{a,b}$ (and corresponding plain bond dimensions $D_{a,b})$ of the optimized tensors for various $D^{*}$ (i.e., number of kept bond multiplets). Here $S^{\left(m\right)}$ means $m$ multiplets with spin $S$.

Figure 4

Spatial dependence of various correlation functions (symbols) on a log-linear scale, together with exponential fits $y=cexp(-x/\xi )$, with $\xi$ indicated with each line. The correlation functions are calculated by iPEPS. $x$ is the distance between triangles with length unit $a_x$ [see Fig. 3]. Note that the square of the converged $\langle S_i^z{S}_{i+1}^z\rangle \ne 0$ has been subtracted in the definition of dimer-dimer correlations.

Figure 5

(a) Energy difference between $A$ and $B$ triangles, $\Delta E=2(E_A-E_B)/3$, where $E_{A\left(B\right)}=9{\langle S_i^z{S}_{i+1}^z\rangle }_{A\left(B\right)}$, evaluated at the Heisenberg point $\left(\theta =0\right)$ with the iPEPS contraction, and plotted vs $1/D$, which show clearly a nonvanishing value $(\delta =\Delta E/e_0\approx 19\%$ for $D=24)$. The minimal bond dimension needed to capture the SVBC order is $D\gtrsim 7$ [or $D^{*}=3$; see the table in Fig. 3]. For smaller $D,\Delta E$ vanishes, and hence is not shown here. The inset shows that $\Delta E$ vanishes when $\theta <-0.04\pi$, where the ferromagnetic quadrupolar order $\left(Q_1\right)$ sets in. (b) Ground-state phase diagram of the spin-1 BLBQ model on the kagome lattice.

Figure 6

(a) Energy per site and (b) von Neumann entanglement entropies of the tensor-network variational wave functions on cylinders. The X(Y)C geometry is shown in Fig. 3, and $L=2,4,6$ means infinite X(Y)C4, 8, 12 cylinders, respectively.