Bulk and edge dynamics of a two-dimensional Affleck-Kennedy-Lieb-Tasaki model

We study the dynamical properties of both bulk and edge spins of a two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model mainly by using the stochastic series expansion quantum Monte Carlo method with stochastic analytic continuation. In the deep AKLT phase, we obtain a spin spectrum with the flat band, which is a strong evidence for a localized state. Through the spectrum analysis, we see a clear continuous phase transition from the AKLT phase to the Neel phase in the model, and the energy gap becomes closed at the corresponding momentum point. In comparison with linear spin-wave theory, we find that there are strong interactions among magnons at high energies. With an open boundary condition, the gap of edge spins in the AKLT phase closes at both the Gamma point and the pi point interestingly to emerge into a flat-band-like Luttinger liquid phase, which can be explained by symmetry and perturbation approximation. This paper helps us to better understand the completely different dynamical behaviors of bulk and edge spins in the symmetry protected topological phase.