Symmetry fractionalization in the topological phase of the spin-$\frac{1}{2}$ $J_1\text{−}{J}_2$ triangular Heisenberg model

Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}{J}_1\text{−}{J}_2$ Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, $Z_2$ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.

Figure 1

Visualization of the triangular lattice on an infinite YC structure. The size and color of the spheres indicate the long-range correlation with the principal (gray) site. The color of the bonds indicates the strength of the NN correlations. The average of NN correlation is subtracted from each bond to highlight the anisotropy pattern.

Figure 2

Ground-state energies at $J_2=0.125$ against inverse cylinder width. Our results for infinitely long cylinders with extrapolation to zero variance are in blue. Dashed lines are guides to the eye. Red shaded symbols are finite-size DMRG results from Ref. [21]. Brown symbols are variational quantum Monte Carlo (VQMC) results on $L\times L$ tori with the horizontal line indicating an $L\to \infty$ extrapolation, from Ref. [22].

Figure 3

The momentum-resolved ES of the even-boundary topological sectors for different cylinder circumferences $L_y$, in the spin-liquid region at $J_2=0.125$. The topological sectors are (a) YC6-$\widehat{\mathrm{v}}$, (b) YC8-$\widehat{\mathrm{i}}$, and (c) YC10-$\widehat{\mathrm{i}}$.

Figure 4

The momentum-resolved ES of the odd-boundary topological sectors for different cylinder circumferences $L_y$, in the spin-liquid region at $J_2=0.125$. The topological sectors are (a) YC6-$\widehat{\mathrm{b}}$, (b) YC8-$\widehat{\mathrm{f}}$, and (c) YC10-$\widehat{\mathrm{b}}$.