The simplest model is the spin-1/2 chain with Majumdar–Ghosh interaction:
H=∑iP3/2(i−1,i,i+1),
where
P3/2(i,j,k) is the projection operator that projects a state onto the subspace with total spin-3/2 on sites
i,j,k. The ground states are two dimer states (see the figure on wikipedia
Majumdar–Ghosh model):
|ψ1⟩=∏i|singlet⟩2i,2i+1,
|ψ2⟩=∏i|singlet⟩2i−1,2i.
If we define the symmetry transformation U(i,j)=exp(iaijP0(i,j)) where P0(i,j) is the singlet projection operator, then U(2i,2i+1)|ψ1⟩=exp(ia2i,2i+1)|ψ1⟩,
U(2i−1,2i)|ψ2⟩=exp(ia2i−1,2i)|ψ2⟩,
for any
i. In other words,
|ψ1⟩ supports a one dimensional representation of the group
U(2i,2i+1) (any
i) which is not a symmetry of the original Hamiltonian. Similar for
|ψ2⟩. It is exactly those emergent symmetries that make this model soluble.
More sophisticated examples can be found here: 0207106.
This post imported from StackExchange Physics at 2014-03-09 08:46 (UCT), posted by SE-user Tengen