Student Seminar: Classical and Quantum Integrable Systems

We will call the first element “spin up” and the second one “spin down”. We introduce
the spin algebra which is generated by the spin variables S α n, where α = 1, 2, 3, with
commutation relations
[S α m, S β n] = iɛ αβγ S γ nδ mn .
The spin operators have the following realization in terms of the standard Pauli
matrices: S α n = 2 σα and the form the Lie algebra su(2). Spin variables are subject
to the periodic boundary condition S α n ≡ S α n+L .
Spin chain. A state of the spin chain can be represented as |ψ〉 = | ↑↑↓↑ · · · ↓↑〉
The Hilbert space of the model has a dimension 2 L and it is
H =
L∏
⊗V n = V 1 ⊗ · · · ⊗ V L
n=1
This space carries a representation of the global spin algebra whose generators are
S α =
L∑
I ⊗ · · · ⊗ Sn
α }{{}
⊗ · · · ⊗ I .
n−th place
n=1
The Hamiltonian of the model is
H = −J
L∑
SnS α n+1 α ,
where J is the coupling constant. More general Hamiltonian of the form
H = −J
n=1
L∑
J α SnS α n+1 α ,
n=1
where all three constants J α are different defines the so-called XYZ model. In what
follows we consider only XXX model. The basic problem we would like to solve is to
find the spectrum of the Hamiltonian H.
– 57 –