We investigate the properties of magnon edge states in a ferromagnetic honeycomb lattice with armchair boundaries. In contrast with fermionic graphene, we find novel edge states due to the missing bonds along the boundary sites. After introducing an external on-site potential at the outermost sites we find that the energy spectra of the edge states are tunable. Additionally, when a non-trivial gap is induced, we find that some of the edge states are topologically protected and also tunable. Our results may explain the origin of the novel edge states recently observed in photonic lattices. We also discuss the behavior of these edge states for further experimental confirmations.Introduction.-One intriguing aspect of electrons moving in finite-sized honeycomb lattices is the presence of edge states, which have strong implications in the electronic properties and play an essential role in the electronic transport [1][2][3]. It is well known that natural graphene exhibits edge states under some particular boundaries [4,5]. For example, there are flat edge states connecting the two Dirac points in a lattice with zig-zag [1] or bearded edges [6]. On the contrary, there are no edge states in a lattice with armchair boundary [7], unless a boundary potential is applied [8].The edge states have also been studied in magnetic insulators [9][10][11], where the spin moments are carried by magnons. Recently, it has been shown that the magnonic equivalence for the Kane-Mele-Haldane model is a ferromagnetic Heisenberg Hamiltonian with the DzialozinskiiMoriya interaction [12,13]. Firstly, while the energy band structure of the magnons of ferromagnets on the honeycomb lattice closely resembles that of the fermionic graphene [14,15], it is not clear whether or not they show similar edge states, particularly in view of the interaction terms in the bosonic models which are usually ignored in graphene [16]. Secondly, most recent experiments in photonic lattices have observed novel edge states in honeycomb lattices with bearded [17] and armchair [18] boundaries, which are not present in fermionic graphene. The main purpose of this paper is to address these two issues. By considering a ferromagnetic honeycomb lattice with armchair boundaries, we find that the bosonic nature of the Hamiltonian reveals novel edge states which are not present in their fermionic counterpart. After introducing an external on-site potential at the outermost sites, we find that the edge states are tunable. Interestingly, we find that the nature of such edge states is , in contrast with the equivalent model for the armchair graphene [8] but, as mentioned earlier, in agreement with the experiments in photonic lattices [17,18]. Furthermore, after introducing a DzialozinskiiMoriya interaction (DMI), we find that the topologically protected edge states are sensitive to the presence of the Tamm-like states and they also become tunable.
