Gapped spin liquid with ${\mathbb{Z}}_2$ topological order for the kagome Heisenberg model

We apply the symmetric tensor network state (TNS) to study the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Our method keeps track of the global and gauge symmetries in the TNS update procedure and in tensor renormalization group (TRG) calculations. We also introduce a very sensitive probe for the gap of the ground state—the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of the Heisenberg model on the kagome lattice is a gapped spin liquid with the ${\mathbb{Z}}_2$ topological order (or toric code type), which has a long correlation length $\xi \sim 10$ unit cells. We justify that the TRG method can handle very large systems with thousands of spins. Such a long $\xi$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than the length of 10 unit cells. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.

Figure 1

(a) TNS on a kagome lattice. Darker purple tensors are site tensors $M^{s_1,s_2,s_3}$ carrying physical spins, while lighter purple $T_{1,2}$ are joint tensors. (b) Simple update for the TNS in which we use the HOOI method [80, 81] to truncate $exp(-\tau H_{▵})T_1{M}^{s_1}{M}^{s_2}{M}^{s_2}$ back to $T_1$ and $M^{s_1,s_2,s_3}$. Similarly for downward triangles. (c) Symmetry condition for the local tensors. Green tensors are the symmetry matrices $G^i$ in Eq. (4). Here “id” stands for identity.

Figure 2

TRG diagrammatic scheme. The original honeycomb lattice can be taken as a triangular lattice in which the unit cell is the six-leg tensor $\mathcal{T}\equiv \mathrm{tTr}\left({\mathbb{T}}_1{\mathbb{T}}_2{\mathbb{T}}_1{\mathbb{T}}_2{\mathbb{T}}_1{\mathbb{T}}_2\right)$ as shown on the left. The triangular lattice TN can be deformed into a TN on a honeycomb lattice [93].

Figure 3

Modular matrices. After every step of TRG, we contract ${\mathbb{T}}_{1,2}$ into ${\mathbb{T}}_1{\mathbb{T}}_2$ and put it on the torus as described by the dashed squares. Due to $\mathrm{SU}\left(2\right)$ symmetry, the result of contracting a single ${\mathbb{T}}_1{\mathbb{T}}_2$ is zero. So we need four ${\mathbb{T}}_1{\mathbb{T}}_2$ tensors to evaluate modular matrices.

Figure 4

The universal ratio $Q$ [Eq. (13)] vs number of TRG steps $N_{\text{RG}}$ for a toric code TNS at the critical point [21]. The system has $2^{N_{\text{RG}}}$ spins after $N_{\text{RG}}$ steps of the TRG iteration.

Figure 5

The universal ratio $Q$ [Eq. (13)] vs number of TRG steps $N_{\text{RG}}$ for our SU(2)-symmetric TNS for the kagome Heisenberg model. The system has $4\times 3^{N_{\text{RG}}+1}$ spins after $N_{\text{RG}}$ steps of the TRG iteration.

Figure 6

Polynomial fitting of the free energy $f\left(\tau \right)$. $D^{*}=12$ and truncation multiplet dimension is ${\chi }^{*}=650$ where the corresponding truncated bond dimension $\chi$ is larger than 1000. The linear coefficient is the variational energy $e_0=-0.4369$. All the fitting coefficients are accurate within a 95% confidence interval.

Figure 7

Variational energy of $S=1/2$ KHAFM for different multiplet dimensions $D^{*}$. The upper bounds on the ground-state energy obtained by the multiscale entanglement renormalization ansatz (MERA) [57], the variational energies obtained by series-expansion methods [54], DMRG [66, 68], Lanczos improved variational Monte Carlo [62], coupled-cluster expansion [65], and nonsymmetric TNS [63] are also shown for comparison.