AKLT model
In
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
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 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).[10]
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.
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:[10]
where is a
Generalizations and extensions
The AKLT model has been solved on lattices of higher dimension,
References
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