AKLT model

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.

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

${\displaystyle {\hat {H}}=\sum _{\langle ij\rangle }{\textit {P}}_{\langle ij\rangle }^{(2)}\sim \sum _{j}{\vec {S}}_{j}\cdot {\vec {S}}_{j+1}+{\frac {1}{3}}({\vec {S}}_{j}\cdot {\vec {S}}_{j+1})^{2}}$

up to a constant, where the ${\textstyle {\vec {S_{i}}}}$ are spin-1 operators, and ${\textstyle {\textit {P}}_{\langle ij\rangle }^{(2)}}$ 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.

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 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 ${\displaystyle S^{z}}$ operator).[10]

The AKLT model has been solved on lattices of higher dimension,[1][15] even in quasicrystals . 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).[20]