Symmetric Antiferromagnetic Hamiltonians

---

Studying symmetric AFM Hamiltonians have triumphed in past decades, most notably Heisenberg model, AKLT model, and so on. This note would try to treat AKLT model from the symmetry perspective.

Let me have some quick questions first.

1. Since the symmetry is enforced by the dot product , can this be generalized to for example the ?
2. How is the coefficient found? (Answered by the Projection construction)

Ref:AKLT on zhihu

Adding a next-nearest neighbor interaction to the Heisenberg model, we have the Majumdar-Ghosh model.

We can write it to

So the ground state should be the state with including the -th sites.

This is a Clebsch-Gordan decomposition of the representation.

Consider the angular momentum addition of a 2-spin system and another spin, the former has, , giving rise to the total .

Oh now it can be found in the table with and , and , .

Note there's also case

No, given by its translation property the triplet state can't be satisfied by nearby double spin-1/2 system. a single translation will give:

So the ground state is

Applying spin inversion symmetry

Consider two spin-1 addition, which has total spin . We can use Casimir operator to classify the irreducible representations.

To project into the

Look at the term which projects into the state, give a positive energy contribution, we would effectively project this out.

Here's the clever point, we can divide the spin-1 to two spin-1/2, and let the nearest neighbor states form singlet. Thus, the onsite pair total spin is restricted to .

In the neighbor spin-1/2 pair, we have

onsite

The AKLT wavefunction:

This is a VBS state, however existing in spin-1 system, and exactly solvable.