The proposal and exact solution of this model by Ian Affleck, Elliott H. Lieb, Tom Kennedy and Hal Tasaki [ja][1] provided crucial insight into the physics of the spin-1 Heisenberg chain.
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.
Affleck et al. were interested in constructing a one-dimensional state with a valence bond between every pair of sites.
Here the solid points represent spin 1/2s which are put into singlet states.
The lines connecting the spin 1/2s are the valence bonds indicating the pattern of singlets.
[10] For the case of spins arranged in a ring (periodic boundary conditions) the AKLT construction yields a unique ground state.
[11] 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.
[12] 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.
[citation needed] The model has also been constructed for higher Lie algebras including SU(n),[16][17] SO(n),[18] Sp(n)[19] and extended to the quantum groups SUq(n).