The AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Ian Affleck, Elliott H. Lieb, Tom Kennedy and provided crucial insight into the physics of the spin-1 Heisenberg chain. It has also served as a useful example for such concepts as valence bond solid order, symmetry-protected topological order and matrix product state wavefunctions. Background A major motivation for the AKLT model was the Majumdar–Ghosh chain. Because two out of every set of three neighboring spins in a Majumdar–Ghosh ground state are paired into a singlet, or valence bond, the three spins together can never be found to be in a spin 3/2 state. In fact, the Majumdar–Ghosh Hamiltonian is nothing but the sum of all projectors of three neighboring spins onto a 3/2 state. The main insight of the AKLT paper was that this construction could be generalized to obtain exactly solvable models for spin sizes other than 1/2. Just as one end of a valence bond is a spin 1/2, the ends of two valence bonds can be combined into a spin 1, three into a spin 3/2, etc. Definition Affleck et al. were interested in constructing a one-dimensional state with a valence bond between every pair of sites. Because this leads to two spin 1/2s for every site, the result must be the wavefunction of a spin 1 system. For every adjacent pair of the spin 1s, two of the four constituent spin 1/2s are stuck in a total spin zero state. Therefore, each pair of spin 1s is forbidden from being in a combined spin 2 state. By writing this condition as a sum of projectors that favor the spin 2 state of pairs of spin 1s, AKLT arrived at the following Hamiltonian up to a constant, where the are spin-1 operators, and the local 2-point projector that favors the spin 2 state of an adjacent pair of spins. This Hamiltonian is similar to the spin 1, one-dimensional quantum Heisenberg spin model but has an additional "biquadratic" spin interaction term. Ground state By construction, the ground state of the AKLT Hamiltonian is the valence bond solid with a single valence bond connecting every neighboring pair of sites. Pictorially, this may be represented as Here the solid points represent spin 1/2s which are put into singlet states. The lines connecting the spin 1/2s are the valence