Student Seminar: Classical and Quantum Integrable Systems

5.2 Algebraic Bethe Ansatz
Here we will solve the Heisenberg model by employing this time a new method called
the Algebraic Bethe ansatz. This method allows one to reveal the integrable structure
of the model as well as to study its properties in the thermodynamic limit.
Fundamental commutation relation. Suppose we have a periodic chain of length L.
The basic tool of the algebraic Bethe ansatz approach is the so-called Lax operator.
The definition of the Lax operator involves the local “quantum” space V i , which for
the present case is chosen to be a copy of C 2 . The Lax operator L i,a acts in V i ⊗ V a :
Explicitly, it is given by
L i,a (λ) : V i ⊗ V a → V i ⊗ V a .
L i,a (λ) = λI i ⊗ I a + i ∑ α
S α i ⊗ σ α ,
where I i , Si
α act in V i , while the unit I a and the Pauli matrices σ α act in an another
Hilbert space C 2 called “auxiliary”. The parameter λ is called the spectral parameter.
Another way to represent that the Lax operator is to write it as 2 × 2 matrix with
operator coefficients
( )
λ + iS
3
L i,a (λ) = i iS − i
iS + i λ − iSi
3 .
Introducing the permutation operator
P = 1 2
(
I ⊗ I +
3∑ )
σ α ⊗ σ α
α=1
we can write the Lax operator in the alternative form
(
L i,a (λ) = λ − i )
I i,a + iP i,a .
2
The most important property of the Lax operator is the commutation relations between
its entries. Consider two Lax operators, L i,a (λ 1 ) and L i,b (λ 2 ), acting in the
same quantum space but in two different auxiliary spaces. The products of these
two operators L i,a (λ 1 )L i,b (λ 2 ) and L i,b (λ 2 )L i,a (λ 1 ) are defined in the triple tensor
product V i ⊗ V a ⊗ V b . Remarkably, it turns out that these two product are related
by a similarity transformation which acts non-trivially in the tensor product V a ⊗ V b
only. Namely, there exists an intertwining operator R a,b (λ 1 , λ 2 ) = R ab (λ 1 − λ 2 ) such
that the following relation is true
R ab (λ 1 − λ 2 )L ia (λ 1 )L ib (λ 2 ) = L ib (λ 2 )L ia (λ 1 )R ab (λ 1 − λ 2 ) . (5.3)
This intertwining operator is called quantum R-matrix and it has the following explicit
form
R ab = λI ab + iP ab
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