Hexagon-singlet solid ansatz for the spin-1 kagome antiferromagnet

We perform a systematic investigation on the hexagon-singlet solid (HSS) states, which are a class of spin liquid candidates for the spin-1 kagome antiferromagnet. With the Schwinger boson representation, we show that all HSS states have exponentially decaying correlations and can be interpreted as a (special) subset of the resonating Affleck-Kennedy-Lieb-Tasaki (AKLT) loop states. We provide a compact tensor network representation of the HSS states, with which we are able to calculate physical observables efficiently. We find that the HSS states have vanishing topological entanglement entropy, suggesting the absence of intrinsic topological order. We also employ the HSS states to perform a variational study of the spin-1 kagome Heisenberg antiferromagnetic model. When we use a restricted HSS ansatz preserving lattice symmetry, the best variational energy per site is found to be $e_0=-1.3600$. In contrast, when allowing lattice symmetry breaking, we find a trimerized simplex valence-bond crystal with a lower energy, $e_0=-1.3871$.

Figure 1

Schematic plots of (a) the spin-1 HSS ansatz on the kagome lattice and (b) the spin-3/2 HSS ansatz on the honeycomb lattices. Six spin-1/2 virtual particles (red dots) form a spin singlet in each hexagon. In (a), the blue ovals denote projectors mapping two virtual spin-1/2's to a physical spin-1, while in (b), the circles denote projectors mapping three spin-1/2's onto a spin-3/2 space.

Figure 2

Illustration of the spin-1 simplex valence-bond crystal state on a kagome lattice. Two neighboring triangles have different energy expectation values, and the lattice inversion symmetry is broken.

Figure 3

Graphical representation of the hexagon singlets. (a) All dimers are between nearest neighbors, like a resonating benzene ring, $|D_{\pm }\rangle$ is with $+$ $(-)$ sign convention in the superposition. (b) An allowed dimer pattern $|N\rangle$ with longer-range valence bonds in the hexagon. The arrow orientates from the first to the second spins in a singlet, which is anti-symmetric towards permutation of the two constituting spin-1/2's. Notice that these three hexagon singlets are all eigenstates of hexagon-inversion symmetry: $|D_{-}\rangle$ and $|N\rangle$ are odd (eigenvalue $-1)$, while $|D_{+}\rangle$ is even $\left(+1\right)$. Moreover, $|D_{\pm }\rangle$ and $|N\rangle$ are all one-dimensional irreducible representations of the point group $C_{6v}$.

Figure 4

(a) Typical resonating AKLT-loop configuration arising from the benzene construction. The green ellipses denote the valence bonds. When cutting the loop configurations either vertically or at 60 degrees to the vertical (denoted by dashed lines), it is only possible to intersect an even number of valence bonds. (b) A typical RAL configuration, shown for comparison. The RAL state has four virtual particles $0\oplus 1/2$ on each vertex (see more details in Ref. [17]), which is a fully packed equal weight superposition of all possible loops, thus contains some configurations that are prohibited in the HSS state. Periodic boundary conditions are assumed on both horizontal and vertical directions.

Figure 5

(a) Tensor network representation of the HSS ansatz on the kagome lattice. (b) The local projector $P$ (dashed ovals) maps two spin-1/2 virtual particles onto the physical spin-1 space. (c) By fusing two virtual spins, we thus introduce composite virtual particle $0\oplus 1$ on the bond. (d) A projection tensor $M_{(S_1,m_1)(S_2,m_2)(S_3,m_3)}^{(S_{d_1},m_{d_1})(S_{d_2},m_{d_2})(S_{d_3},m_{d_3})}$ $[(S_i,m_i)$'s are geometric indices along red solid lines and $(S_{d_i},m_{d_i})$'s are physical indices], and (e) a rank-three hexagon tensor $R_{(s_1,m_1)(s_2,m_2)(s_3,m_3)}$ are obtained using the composite virtual particles. $S$ and $m$ denote the spin and magnetic quantum numbers, respectively.

Figure 6

The tensor networks on (a) XC and (b) YC cylindrical geometries, where periodic (open) boundary condition in vertical (horizontal) direction is assumed. The length units $a_{x,y}=1$ are also shown.

Figure 7

(a) The spin-spin, quadrupole-quadrupole, and dimer-dimer correlation functions of the ${|\mathrm{BRS}\rangle }_{\mathrm{O}}$, obtained by SU(2) iPEPS contractions. All of the correlation functions $C\left(x\right)$ are found to decay exponentially, as $C\left(x\right)\sim exp(-x/\xi )$. The correlation lengths $\xi$ are obtained from fitting the data. (b) Entanglement entropies of the odd BRS on XC and YC geometries with various circumferences (up to $L=16)$, the entropy data extrapolate to $\gamma =0$ in the $L=0$ limit.

Figure 8

(a) Energy expectation values (per site) $e_0$ of Hida's HSS state on cylinders with various circumferences $L$ [see Fig. 6 for the illustration of the XC and YC geometries, the length units $a_x,a_y=1$ are also shown in Fig. 6]. The energy results converge very fast to the thermodynamic limit obtained by iPEPS contractions. (b) Entanglement entropy results extrapolate to $\gamma \simeq 0$ as $L\to 0$, for both XC and YC geometries.

Figure 9

Variational energies of the HSS states. (a) By adding the hexagon-singlet configurations in Fig. 3, with weight $\omega$, to the benzene-ring state, we connect smoothly the latter $\left(\omega =0\right)$ with the Hida point $\left(\omega =0.582618977\right)$. The lowest variational energy $\left(e_0=-1.359975\right)$ is slightly lower than that of the Hida point. (b) By tuning parameters $\alpha$ and $\beta$, we obtain the lowest energy $e_0\simeq -1.36000$ with $\alpha =-0.2438$ and $\beta =0.134$, which has no energy difference between the two kinds of triangles, i.e., $e_A\simeq e_B$.

Figure 10

Variational study of the spin-1 KHAF model beyond the HSS ansatz. (a) Although the hexagon singlets are intact, applying an imaginary-time evolution operator on the triangle tensor $M$ will mess up the HSS picture, where the three-site triangle operator $I-\tau H_{▿}$ is shown in (b).

Figure 11

(a) Energies per site $e_0$ for fixed $\tau =0.23$. By tuning $\alpha ,\beta$, we find the lowest variational energy $e_0=-1.3871$ is at $\alpha =-0.5$ and $\beta =0.384$, with $e_A-e_B\ne 0$. (b) We collect the curves that possess the lowest variational energy for various fixed $\tau$. The global minimum is on the $\alpha =-0.5$ and $\tau =0.23$ curve, and with $\mathrm{\Delta }e/e_0\sim 20\%[\mathrm{\Delta }e=2(e_A-e_B)/3]$.

Figure 12

The entanglement spectra (ES) on YC geometries, with various cylinder widths $L$. Each symbol in the figure labels a multiplet, instead of an individual plain state. The spectra of different circumferences have been offset so as the lowest eigenvalues are zero. (a) The ES of Hida's HSS state on cylinders with $L=4,6,8,10,12,14$. There exists a nonvanishing singlet-triplet gap $\left(\delta \sim 1.6\right)$ in the spectra. (b) The ES of the $S=2$ square-lattice AKLT state on cylinders with $L=4,6,8,10,12,14$, where the singlet-triplet gaps $\delta$ $\left(\sim 1/L\right)$ extrapolate to zero in the large $L$ limit.