bonds indicating the pattern of singlets. The ovals are projection operators which "tie" together two spin 1/2s into a single spin 1, projecting out the spin 0 or singlet subspace and keeping only the spin 1 or triplet subspace. The symbols "+", "0" and "−" label the standard spin 1 basis states (eigenstates of the operator). Spin 1/2 edge states For the case of spins arranged in a ring (periodic boundary conditions) the AKLT construction yields a unique ground state. But for the case of an open chain, the first and last spin 1 have only a single neighbor, leaving one of their constituent spin 1/2s unpaired. As a result, the ends of the chain behave like free spin 1/2 moments even though the system consists of spin 1s only. The spin 1/2 edge states of the AKLT chain can be observed in a few different ways. For short chains, the edge states mix into a singlet or a triplet giving either a unique ground state or a three-fold multiplet of ground states. For longer chains, the edge states decouple exponentially quickly as a function of chain length leading to a ground state manifold that is four-fold degenerate. By using a numerical method such as DMRG to measure the local magnetization along the chain, it is also possible to see the edge states directly and to show that they can be removed by placing actual spin 1/2s at the ends. It has even proved possible to detect the spin 1/2 edge states in measurements of a quasi-1D magnetic compound containing a small amount of impurities whose role is to break the chains into finite segments. In 2021, a direct spectroscopic signature of spin 1/2 edge states was found in isolated quantum spin chains built out of triangulene, a spin 1 polycyclic aromatic hydrocarbon. Matrix product state representation The simplicity of the AKLT ground state allows it to be represented in compact form as a matrix product state. This is a wavefunction of the form Here the As are a set of three matrices labeled by and the trace comes from assuming periodic boundary conditions. The AKLT ground state wavefunction corresponds to the choice: where is a Pauli matrix. Generalizations and extensions The AKLT model has been solved on lattices of higher dimension, even in quasicrystals . The model has also been constructed for higher Lie algebras including SU(n), SO(n), Sp(n)