Student Seminar: Classical and Quantum Integrable Systems

The monodromy is an operator on V L ⊗ V L−1 ⊗ . . . ⊗ V 1 ⊗ V a . It we take the trace
of the monodromy w.r.t. to its matrix part acting in the auxiliary space we obtain
an object which is called the transfer matrix and it is denoted as τ(λ) = tr a T a (λ).
Denote L = L i,a and L ′ = L i+1,a
R 12 L ′ 1L 1 L ′ 2L 2 = R 12 L ′ 1L ′ 2L 1 L 2 = L ′ 2L ′ 1R 12 L 1 L 2 = L ′ 2L ′ 1L 2 L 1 R 12 = L ′ 2L 2 L ′ 1L 1 R 12 .
This is because L 1 and L ′ 2 commute – they act both in different auxiliary spaces and
different quantum spaces. Thus, we deduce the commutation relation between the
components of the monodromy
R 12 (λ − µ)T 1 (λ)T 2 (µ) = T 2 (µ)T 1 (λ)R 12 (λ − µ) .
Now we can proof the fundamental fact about the commutation relations above.
Rewrite them in the form
T 1 (λ)T 2 (µ) = R 12 (λ − µ) −1 T 2 (µ)T 1 (λ)R 12 (λ − µ)
and takes the trace over the first and the second space. We will get
)
τ(λ)τ(µ) = tr 1,2
(R 12 (λ − µ) −1 T 2 (µ)T 1 (λ)R 12 (λ − µ) = τ(µ)τ(λ) .
Thus, the transfer matrices commute with each other for different values of the
spectral parameter
[τ(λ), τ(µ)] = 0 .
Hence, τ(λ) generates an abelian subalgebra. If we find the Hamiltonian of the model
among this commuting family then we can call our model quantum integrable. The
Hamiltonian must be
H = ∑ d k
c ka
dλ ln τ(λ)| k λ=λ a
.
a,k
for some coefficients c ka . This will ensute that the Hamiltonian belongs to the family
of commuting quantities. Since all the integrals from this family mutually commute
they can be simultaneously diagonalized.
Represent the monodromy as the 2 × 2 matrix in the auxiliary space
( )
A(λ) B(λ)
T (λ) =
,
C(λ) D(λ)
where the entries are operators acting in the space ⊗ L i=1V i . From the definition of
the monodromy and the L-operator it is clear that T is a polynomial in λ and
T (λ) = λ L + iλ L−1
L
∑
n=1
S α n ⊗ σ α + · · ·
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