Student Seminar: Classical and Quantum Integrable Systems

This allows to establish that
dτ(λ)
dλ τ(λ)−1 | λ=i/2 =
= i −1 ( ∑
n
P 12 P 23 · · · P n−1,n+1 · · · P L−1,L
)(
P L,L−1 P L−1,L−2 · · · P 2,1
)
= 1 i
L∑
n,n+1
P n,n+1 .
On the other hand we see that
H = −J
L∑
SnS α n+1 α = − J 4
n=1
L∑
n=1
( 1
σnσ α n+1 α = −J
2
L∑
P n,n+1 − L 4
n=1
)
.
Hence,
( i dτ(λ)
H = −J
2
)
| λ=i/2 ,
dλ τ(λ)−1 − L 4
i.e. the Hamiltonian belongs to the family of L − 1 commuting integrals. To obtain
L commuting integrals we can add the operator S 3 to this family.
The spectrum of the Heisenberg model. Here we compute the eigenvalues of H by
using the algebraic Bethe ansatz. First we derive the commutation relations between
the operators A, B, C, D. The form of the R-matrix is
⎛
⎞
λ − µ + i 0 0 0
0 λ − µ i 0
R(λ − µ) = ⎜
⎟
⎝ 0 i λ − µ 0 ⎠ .
0 0 0 λ − µ + i
We compute
⎛
⎞
⎛
⎞
A(λ) 0 B(λ) 0
A(µ) B(µ) 0 0
0 A(λ) 0 B(λ)
T a (λ) = ⎜
⎟
⎝ C(λ) 0 D(λ) 0 ⎠ , T C(µ) D(µ) 0 0
b(λ) = ⎜
⎟
⎝ 0 0 A(µ) B(µ) ⎠ .
0 C(λ) 0 D(λ)
0 0 C(µ) D(µ)
Plugging this into the fundamental commutation relation we get
⎛
⎞
(α + i)A λ A µ (α + i)A λ B µ (α + i)B λ A µ (α + i)B λ B µ
αA λ C µ + iC λ A µ αA λ D µ + iC λ B µ αB λ C µ + iD λ A µ αB λ D µ + iD λ B µ
⎜
⎟
⎝ iA λ C µ + αC λ A µ iA λ D µ + αC λ B µ iB λ C µ + αD λ A µ iB λ D µ + αD λ B µ ⎠ =
(α + i)C λ C µ (α + i)C λ D µ (α + i)D λ C µ (α + i)D λ D µ
⎛
⎞
(α + i)A µ A λ αB µ A λ + iA µ B λ iB µ A λ + αA µ B λ (α + i)B µ B λ
(α + i)C
=
µ A λ αD µ A λ + iC µ B λ iD µ A λ + αC µ B λ (α + i)D µ B λ
⎜
⎟
⎝ (α + i)A µ C λ αB µ C λ + iA µ D λ iB µ C λ + αA µ D λ (α + i)B µ D λ ⎠ .
(α + i)C µ C λ αD µ C λ + iC µ D λ iD µ C λ + αC µ D λ (α + i)D µ D λ
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