## 3 Numerical Results
### Vortex Solitons  
Unlike multipole solitons, vortex solitons have more complicated phase structures and carry angular momenta. The main difference between the multipole solitons and vortex solitons is their phase structures. For the multipole solitons that are discussed above, their phase structures are in-phase. For vortex solitons, their phase structures depend on their topological charges, \(m\). Moreover, the sign of \(m\) denotes the direction of the angular momentum (counterclockwise for \(m>0\), clockwise for \(m<0\)). We find that the PT-symmetric triangular lattices with nonlocal nonlinearity can support vortex solitons with different topological charges. To compare the phase structures of the vortex solitons, we choose three solitons with \(m=1,2,3\), as shown in Figure 10. Their other parameters are the same, i.e., \(d=0.5,\mu=10\). They have the same profile as Figure 10a, which is composed of six humps. All their phase structures have gradual phase differences in a spiral form. In Figure 10b,d, we show the phase structures of these vortex solitons. For the vortex soliton with unity topological charge (\(m=1\)), its phase structure is restored to its original shape after rotating by \(2\pi\). For a vortex soliton \(m=2\), the phase structure is restored to its original shape after \(\pi\) of rotation. As for the one with \(m=3\), the angle of phase structure recovery is \(2\pi/3\). The circles in Figure 10 denote the positions of the soliton humps. During the propagation of these vortex solitons, their phase structures rotate counterclockwise. We take the vortex solitons with \(m=3,d=0.5,\mu=9\) as an example, which is shown in Figure 11.  
Figure 10: The profile and the phase structures of vortex solitons with different topological charges. (**a**) is their profile \(|U|^{2}\), i.e., square of modules of soliton solution \(U\). (**b–d**) correspond to the phase structures of vortex solitons with \(m=1,2,3\). \(m=\) topological charges. According to numerical calculations, the vortex solitons can exist in both the first bandgap (\(4.82<\mu<10.67\)) and the infinite bandgap (\(\mu>10.75\)). For the multipole solitons discussed above, the stable solitons have the same profiles (dipole or six-pole). For vortex solitons, however, their profiles are influenced by the propagation constant, \(\mu\), the degree of nonlocality, \(d\), and the topological charges, \(m\). These vortex solitons can maintain their initial profiles during the propagation past the point where \(z=140\). To illustrate this novel property, we take vortex solitons with \(m=2\) or \(m=3\) and with different \(\mu\) as an example. The profiles of the vortex solitons under intermediate nonlocal nonlinearity (\(d=0.5\)) with \(m=3\) are shown in Figure 12. The upper row corresponds to the vortex solitons in the first bandgap, and the lower row corresponds to the infinite bandgap. The shape of these vortex solitons should be a six-pole profile in which the centers of each pole are arranged symmetrically around a complete circle. However, as these solitons are coming from far from the edge of the bandgap (\(\mu=10.67\) for the first bandgap and \(\mu=10.75\) for the infinite bandgap), their profiles cannot form a complete circle. We will call this \"incomplete\". In detail, the number of the poles decreases as \(\mu\) is far from the edge, which indicates that \(\mu\) greatly affects the profiles of vortex solitons. Moreover, the profiles of the vortex solitons with larger \(d\) tend to be incomplete. Figure 11: The phase structure evolution of vortex solitons with \(m=3,d=0.5,\mu=9\). (**a–c**) correspond to the phase structures of vortex solitons with \(z=0.25,1.2,2.15\). The arrows point to the phases of one specific peak. Figure 12: The evolution of the profiles of vortex solitons with \(m=3,d=0.5\). (**a–f**) are the profiles of the vortex solitons with \(\mu=8.2,8.4,10.6,11,11.4,12.2\), respectively. The upper row corresponds to solitons in the first bandgap, while the lower row corresponds to solitons in the infinite bandgap.