Characterizing $S=3/2$ AKLT Hamiltonian with Scanning Tunneling Spectroscopy

The AKLT Hamiltonian is a particular instance of a general class of model Hamiltonians defined in lattices with coordination $z$ where each site hosts a spins $S=z/2$, interacting both with linear and non-linear exchange couplings. In two dimensions, the AKLT model features a gap in the spectrum, and its ground state is a valence bond solid state; that is an universal resource for measurement based quantum computing, motivating the quest of physical systems that realize this Hamiltonian. Given a finite-size system described with a specific instance of this general class of models, we address the question of how to asses if such system is a realization of the AKLT model using inelastic tunnel spectroscopy implemented with scanning tunnel microscopy (IETS-STM). We propose two approaches. First, in the case of a dimer, we show how to leverage non-equilibrium IETS-STM to obtain the energies of all excited states, and determine thereby the magnitude of both linear and non-linear exchange interactions. Second, we explore how IETS can probe the in-gap excitations associated to edge spins. In the AKLT limit, spins $S=3/2$ at the edge of the lattice have coordination 2, giving rise to $S=1/2$ dangling spins that can be probed with IETS. We propose a $S=1/2$ effective Hamiltonian to describe the interactions between these dangling spins in the neighborhood of the AKLT point, where their degeneracy lifted.